How to glue derived categories
Abstract
We give an overview of existing enhancement techniques for derived and trianguated categories based on the notion of a stable model category, and show how it can be applied to the problem of gluing triangulated categories. The article is mostly expository, but we do prove some new results concerning existence of model structures.
Keywords
Derived category Triangulated category Gluing1 Introduction and historical overview
The basic question addressed in this paper is formulated in the title, and it looks extremely oldfashioned: while in the early 1990ies, this would have been a hot research topic, by now so many solutions are available in the literature that the question looks completely closed. To explain why in our view the question is still relevant, let us start with a brief historical overview.
1.1 Derived categories
From its very beginning [10, 20], homological algebra was based on the notion of a right and leftderived functor. For rightderived functors, one starts with a leftexact functor \(F:\mathcal {C}\rightarrow \mathcal {E}\) between abelian categories \(\mathcal {C}\), \(\mathcal {E}\). To construct the rightderived functors \(R^\cdot F\), one replaces an object \(c \in \mathcal {C}\) by an appropriate (for example, injective) resolution, applies F, and takes cohomology. For leftderived functors, the definition is dual. By a general theorem, the result of the procedure does not depend on the choice of a resolution.
The formalism of derived categories that appeared in [46] internalizes this independence: to define the derived category \({\mathcal {D}}(\mathcal {C})\), one takes the category \(C_\cdot (\mathcal {C})\) of chain complexes of objects in \(\mathcal {C}\), and formally inverts quasiisomorphisms. Since two resolutions of the same object are quasiisomorphic, they give the same object in \({\mathcal {D}}(\mathcal {C})\). Therefore \({\mathcal {D}}(\mathcal {C})\) is the natural domain of definition for derived functors.
The procedure of formally inverting a class of maps in a category is known as localization, and it is rather nontrivial. For \(C_\cdot (\mathcal {C})\), one does it in two steps. First, one considers the category \(H(\mathcal {C})\) whose objects are chain complexes in \(\mathcal {C}\), and whose maps are chainhomotopy equivalence classes of maps in \(C_\cdot (\mathcal {C})\). Then one localizes \(H(\mathcal {C})\) to obtain \({\mathcal {D}}(\mathcal {C})\). For the second step, one uses an additional structure carried by \(H(\mathcal {C})\) and \({\mathcal {D}}(\mathcal {C})\)—that of a triangulated category. This notion also appeared in [46], following an earlier version in [12]. The crucial difference is the socalled octahedron axiom missing in [12] that allows one to prove the Verdier Localization Theorem: given a full triangulated subcategory \({\mathcal {D}}' \subset {\mathcal {D}}\) in a triangulated category \({\mathcal {D}}\) satisfying some mild conditions, one can construct a triangulated “quotient category” \({\mathcal {D}}/{\mathcal {D}}'\) that is the localization of \({\mathcal {D}}\) with respect to all maps whose cone lies in \({\mathcal {D}}'\). To obtain \({\mathcal {D}}(\mathcal {C})\), one takes \({\mathcal {D}}=H(\mathcal {C})\), and lets \({\mathcal {D}}'\) be the full subcategory spanned by acyclic complexes.
1.2 Triangulated categories

Ease of construction. Any condition on objects in a triangulated category \({\mathcal {D}}\) closed under taking cones defines a full triangulated subcategory \({\mathcal {D}}' \subset {\mathcal {D}}\); if needed, one can then take the quotient \({\mathcal {D}}/{\mathcal {D}}'\), find another full subcategory inside it, and so on.

Ease of comparison. For example, assume given triangulated categories \({\mathcal {D}}\), \({\mathcal {D}}'\) with generators \(E \in {\mathcal {D}}\), \(E' \in {\mathcal {D}}'\), and a triangulated functor \(F:{\mathcal {D}}' \rightarrow {\mathcal {D}}\) such that \(F(E')=E\). Then to check that F is an equivalence, it suffices to prove that the induced map \(F:{\text {Hom}}(E',E'[n]) \rightarrow {\text {Hom}}(E,E[n])\) is an isomorphism for any integer n.
However, it became clear very soon that the notion of a triangulated category has grave deficiencies: several important constructions that work for abelian categories stop working in the triangulated setting. For example, if we have a leftexact functor \(F:\mathcal {C}\rightarrow \mathcal {C}'\) between abelian categories, then the category \(\widetilde{\mathcal {C}}\) of triples \(\langle c,c',f \rangle \), \(c \in \mathcal {C}\), \(c' \in \mathcal {C}'\), \(f:c' \rightarrow F(c)\) is abelian (this is known as gluing). The corresponding statement for triangulated categories is not true. Worse than that, there is no way to recover \({\mathcal {D}}(\widetilde{\mathcal {C}})\) from \({\mathcal {D}}(\mathcal {C})\), \({\mathcal {D}}(\mathcal {C}')\) and the derived functor \(R^\cdot F:{\mathcal {D}}(\mathcal {C}) \rightarrow {\mathcal {D}}(\mathcal {C}')\).
More generally, the notion of a family of categories parametrized by a small category I is conveniently formalized as a fibration \(\mathcal {C}\rightarrow I\) in the sense of [21] (a Grothendieck fibration). Then if the fibers \(\mathcal {C}_i\), \(i \in I\) of the fibration are abelian, and the transition functors are leftexact, the category \({\text {Sec}}(\mathcal {C})\) of sections \(I \rightarrow \mathcal {C}\) of the fibration \(\mathcal {C}\rightarrow I\) is abelian. For triangulated categories, the statement is wrong, and moreover, one cannot recover the derived category \({\mathcal {D}}({\text {Sec}}(\mathcal {C}))\) from the family of the derived categories \({\mathcal {D}}(\mathcal {C}_i)\), \(i \in I\). This happens even when the family is the constant family \(\mathcal {C}= \mathcal {C}_0 \times I\), so that \({\text {Sec}}(\mathcal {C})\) is the category \(\mathcal {C}_0^I\) of functors from I to an abelian category \(\mathcal {C}_0\): one cannot recover \({\mathcal {D}}(\mathcal {C}_0^I)\) from I and \({\mathcal {D}}(\mathcal {C}_0)\).
One can say that the structure of a triangulated category captures some of the natural structure possessed by a derived category \({\mathcal {D}}(\mathcal {C})\) but not all of it; a satisfactory notion would be a triangulated category “with enhancement”.
Unfortunately, finding out what this “enhancement” might be, exactly, turned out to be an extremely difficult problem.
1.3 Enhancements
Historically, the first notion of an enhancement was suggested by Grothendieck [23], based on his idea of a derivator. This is very pleasing conceptually but has technical problems; although it has been vigorously developed, it is even now perhaps not ready for practical use.
A more practical alternative is the socalled DG enhancement. For details on this, we refer the reader to [30] or to a comprehensive recent overview in [38, Section 2]. Here is the main idea. In the derived category case, one observes for any two objects \(c,c' \in C_\cdot (\mathcal {C})\), we have a whole complex of morphisms \({\text {Hom}}^\cdot (c,c')\). Maps in \(H(\mathcal {C})\) correspond to degree0 cohomology classes of this complex. One axiomatizes the situation by introducing a notion of a DG category \(\mathcal {C}_\cdot \) that by definition has a complex \(C_\cdot (c,c')\) of morphisms for any two objects. Taking degree0 cohomology of these complexes, we obtain a category \(H^0(\mathcal {C}_\cdot )\). One distinguishes a class of pretriangulated DG categories \(\mathcal {C}_\cdot \) for which \(H^0(\mathcal {C}_\cdot )\) has a natural triangulated structure. One says that a DG functor \(F:\mathcal {C}_\cdot \rightarrow \mathcal {C}_\cdot '\) is a quasiequivalence if \(F:\mathcal {C}_\cdot (c,c') \rightarrow \mathcal {C}'_\cdot (F(c),F(c')\) is a quasiisomorphism for any \(c,c' \in \mathcal {C}\), and the induced functor \(H^0(F):H^0(\mathcal {C}_\cdot ) \rightarrow H^0(\mathcal {C}'_\cdot )\) is an equivalence. Then a DG enhancement of a triangulated category \({\mathcal {D}}\) is a pretriangulated DG category \({\mathcal {D}}_\cdot \) equipped with an equivalence \(H^0({\mathcal {D}}_\cdot ) \cong {\mathcal {D}}\), considered up to a quasiequivalence.
At a first glance, this approach feels like a throwback to earlier times when one had to choose resolutions, with the only difference that now we choose all resolutions at the very beginning. Independence of this choice is now encoded in the fact that we consider DG categories up to a quasiequivalence. This is again a localization problem: we have the category of DG categories, and we want to invert quasiequivalences between them.
Since the category of DG categories is not additive, this localization problem requires new methods. Fortunately, these methods do exist, in the form of model structures and model categories introduced by Quillen [39].
Model categories were somewhat neglected for a while, but the subject has experienced a real resurgence starting approximately from the book [26], and has been very active since then. In particular, a good model structure on the category of DG categories has been constructed in [43], and the corresponding localized category \(\mathsf{dgcat}\) has been studied extensively in [44]. This gives precise meaning to the phrase “DG category up to a quasiequivalence”, and makes DG enhancements a workable tool.
 (i)
It only work fors “algebraic” triangulated categories in the sense of [30]; there are many interesting categories that come from algebraic topology, and these are known not to admit DG enhancements (see e.g. [42]).
 (ii)
A DG category is not a category—it is a category with an additional structure. Therefore we cannot package a family \(\mathcal {C}_\cdot \) of DG categories parametrized by some I into a single category by the Grothendieck construction. Moreover, even if we forget the additional structure, the corresponding fibered category \(\mathcal {C}\rightarrow I\) itself is not a DG category, and neither is the category of sections \({\text {Sec}}(\mathcal {C})\). In particular, the category \(\mathcal {C}^I\) of functors from a small category I to a DG category \(\mathcal {C}\) is not a DG category. Both \({\text {Sec}}(\mathcal {C})\) and \(\mathcal {C}^I\) can be made into DG categories by additional technical trickery, but the construction is unpleasant and depends on irrelevant choices.
1.4 Recent developments
Recent resurgence of interest in the model categories is actually explained by developments in algebraic topology, where the late 1990ies saw a tremendous push to put the foundations of the subject on a more solid ground, with remarkable achievements such as “brave new algebra” of [18, 27]. It seems that at first, the notion of a triangulated category was not considered very useful in topology. However, the situation corrected itself very soon, and many versions of enhancements appropriate in the topological context appeared in the literature.
It is certainly too early to summarize this very active area. However, it probably would not be too far off the mark to say that most recent developments focus on the notion of an \(\infty \) category.
As an idea, this goes back to [23], and one way to phrase it is the following. What localization produces naturally is not just a category, it is in fact a category enriched in homotopy types: for any two objects c, \(c'\), we have a homotopy type \({\text {Hom}}(c,c')\) whose \(\pi _0\) is the set of maps in the localized category (for a precise construction of \({\text {Hom}}(c,c')\) see [15]). If one wants to also keep track of \(\pi _1\), then up to an equivalence, a homotopy type can be replaced with its Poincaré groupoid. A category enriched in groupoids is a 2category, or rather, a (2, 1)category (all 1morphisms are invertible). What one would like to have, then, are notions of an “\(\infty \)groupoid” and an “\((\infty ,1)\)category”—higher versions of groupoids and (2, 1)categories that keep track of all the homotopy groups. This is what localization should produce, and this is the correct structure to consider. One can further distinguish a subclass of stable \(\infty \)categories; these are automatically enriched in stable homotopy types, produce triangulated categories after truncation, and give the correct notion of enhancement.
 (i)
It was felt that it certainly can be done, and in several ways.
 (ii)
All of the constructions would be very complicated and ad hoc. There is no preferred construction.
 (iii)
This would not matter: all of the constructions will be equivalent, and moreover, “the space of all possible constructions is contractible”.
 (iv)
Moreover, one could even turn the last statement into a theorem.
 (v)
However, to do so, one needs to make sense of “the space of all constructions”. This is certainly possible, and there are several ways to do it, just as in (i). But then one also has (ii), (iii), (iv), (v), and the infinite loop.
In a way, it is also free from the second problem, but this comes at a price. Effectively, one observes that a category is also a \(\infty \)category, and a family \(\mathcal {C}_\cdot \) of \(\infty \)categories indexed by some I can be turned into a single \(\infty \)functor \(\mathcal {C}\rightarrow I\) by an \(\infty \)version of the Grothendieck construction. Then to glue the family, one can again consider the sections of this functor. However, to get the correct result, these must be \(\infty \)sections. By a general theorem, these exist and form an \(\infty \)category, stable if so where the fibers \(\mathcal {C}_i\), but this is all one can say for free. Getting any sort of control over the result of the procedure requires one to actually go into the definitions, and none of them are easy or constructive.
In practice, people try to use the machinery of \(\infty \)categories as a black box, but the results are somewhat mixed—it is hard to work with a black box that contains not only proofs but also statements and definitions. It seems that the \(\infty \)technology as presently developed is of an allornothing type: either we move the whole of mathematics to the \(\infty \)categorical setting, or we cannot use it at all. This might have been tempting, were the setting really simple or really natural, but it is neither, with a lot of pretty arbitrary choices hardcoded into its very foundations. So, while the move might happen eventually, now is clearly not the time.
 (i)
Why do we have to replace sets with simplicial sets? Chain complexes and cohomology long exact sequences emerge naturally when one considers functors exact on one side, and tries to find obstructions to exactness. The category \(\Delta \) appears quite naturally in algebra and category theory (e.g. [37, VII.5]); the fact that it has something to do with derived functors and enhancements is just postulated. We have a pretty good idea of why simplicial sets give a good model for topological spaces, and vice versa (see e.g. [13] or [32]), so to a topologist, there is no problem; everybody else has to just take it on faith.
 (ii)
Why do we need to replace the naive localization with the Dwyer–Kan localization? The motivation given by Dwyer and Kan is group completion: for a free monoid M, its group completion \(\Omega BM\) is just the free group, and if M is not free, it can have higher homotopy. For a topologist, or at least for someone with some experience of topology, this is quite convincing. For everybody else, this makes no sense: why would one involve \(\Omega BM\), a topological space, in such a purely algebraic question? Tensor product is exact on the right and not on the left, so we have to derive it; an instrinsic explanation of why we have to derive localization is not in the literature.
1.5 The content of the paper
Let us now explain what we do in this paper. We certainly do not attempt to suggest one more general notion of enhancement claimed to be better than all the other ones. Instead, we take a step back and try to see how far one can go with methods already wellestablished in the literature. Our interest is entirely in practical applications, mostly to algebraic geometry and representation theory. We want to show that with some careful assembly and a couple of new insights, already the downtoearth oldfashioned approaches give a reasonably simple and direct notion of enhancement sufficient for many (but not all) practical purposes.
The enhancement we suggest is based on stable model categories, and the fact that these produce enhancements has been known since at least [26], and probably goes back to Goodwillie and/or Franke (informally, it was probably known to Quillen, although since the notion of a stable model category is not in [39], his remark in [39, I, §3, Prop. 4] to this effect makes no sense). In fact, one can use model structures already to define the derived category \({\mathcal {D}}(\mathcal {C})\) of an abelian category \(\mathcal {C}\). This only works if \(\mathcal {C}\) is large enough (for instance, has enough projectives or enough injectives); to compensate for this, we make one small generalization by replacing stable model categories with “stable model pairs” of a model category \(\mathcal {C}'\) and a full subcategory \(\mathcal {C}\subset \mathcal {C}'\) that is closed under weak equivalences and stable in an appropriate sense. Again, the idea that it helps to do this has been around for some time. We give precise definitions in Sect. 2, and we prove that stable model pairs do produce triangulated categories. The proof itself does not appear in the literature in the exact form we need, but it is definitely not new.
In Sect. 3, we give the first application, namely, to an elementary gluing situation. The statement here is not difficult and amounts to one small observation, but we believe that it is new. The main novelty is that we allow gluing functors that do not preserve limits nor colimits, nor even products or coproducts (such as e.g. polynomial functors)—this generality is needed if one wants to move beyond algebraic triangulated categories of [30]. We also show that in the language of triangulated categories, our procedure produces a semiorthogonal decomposition of [6].
We then move to the general gluing situation, and already in the purely categorical setting with no homotopies in sight, it takes some time to work out precisely what this should mean.
We start by attempting to iterate the elementary gluing, and this leads very quickly to the Grothendieck construction of [21]; we recall the corresponding notions briefly in the beginning of Sect. 4. A good notion of a family of categories to be glued is then given by a Grothendieck precofibration \(\mathcal {C}\rightarrow I\) over a small base category I, and one can describe the result of the gluing axiomatically as a certain 2categorical limit. Somewhat surprisingly, this is not the category \({\text {Sec}}(I,\mathcal {C})\) of sections of the precobiration but a different gadget that we call the category of cosections and denote \({\text {Rec}}(I,\mathcal {C})\). We then briefly recall another example of a gluing situation that occurs in nature, namely, that of a comonad \(\Phi \) on a category \(\mathcal {C}\). Here the result of gluing is the category \({\text {Coalg}}(\mathcal {C},\Phi )\) of coalgebras over \(\Phi \).
In Sects. 5 and 6, we give a uniform treatment of the categories \({\text {Rec}}(I,\mathcal {C})\) and \({\text {Coalg}}(\mathcal {C},\Phi )\), by interpreting both as sections of a Grothendieck prefibration on a larger category—the category of simplices \(\Delta \) for \({\text {Coalg}}(\mathcal {C},\Phi )\), and the socalled simplicial replacement \(\Delta I\) of I for \({\text {Rec}}(I,\mathcal {C})\). The main motivation for doing this is that sections are intrinsic and easy to derive, and they package nicely all the higher associativity constraints that emerge in the process. The theory itself is also reasonably complete. The proofs are longish, since we try to do things in a very invariant fashion, but nothing is difficult. Let us also mention that the material in Sects. 4, 5, and 6 is purely categorical and completely modelindependent—were one to try to do an \(\infty \)categorical version of the gluing story, this part would have been exactly the same.
In Sects. 7 and 8, we return to the model structures. The main technology here is the Reedy model structure on the category of functors \(\mathcal {C}^I\) from a small category I to a model category \(\mathcal {C}\). This requires strong assumptions on I but no assumptions on \(\mathcal {C}\). Fortunately, and this crucial observation has been known since the groundbreaking paper [14], the simplicial replacement \(\Delta I\) of an arbitrary small category I is a Reedy category in a natural way. Since we work over simplicial replacements anyway, Reedy structures are a natural tool to use, and all we need to do is to generalize them to the categories of sections of a Grothendieck prefibration. This has been accomplished recently in rather large generality by Balzin [1]; we recall his result in Theorem 7.17. We also briefly recall the construction, since it fits nicely together with our elementary gluing construction of Sect. 3 (roughly speaking, Balzin’s version of the Reedy model structure is obtained by iterating the elementary gluing by induction on degree).
One technical complication occurs because Theorem 7.17 only applies to Reedy categories of a special type, and in order to produce such a category, one needs to do an additional simplicial replacement (we do it in Sect. 7.4 under the name of a “matching expansion”). If one is prepared to restrict oneself to rightexact gluing functors, then this can be avoided, and the proofs in Sects. 7 and 8 can be considerably simplified. In algebraic situations, this is a reasonable thing to do, but since we also want to cover examples such as polynomial gluing functors, we do not do this.
Having finished with all the technical machinery, we finally move to our main subject, namely, gluing of triangulated categories that come from stable model pairs. In Sect. 9, we produce a derived version \({\text {DRec}}(I,\mathcal {C})\) of the category \({\text {Rec}}(I,\mathcal {C})\), and in Sect. 10, we do the same for the category \({\text {Coalg}}(\mathcal {C},\Phi )\). We also construct a useful spectral sequences that computes \({\text {Hom}}\)groups in \({\text {DRec}}(I,\mathcal {C})\), and as an easy application, we show that any triangulated category that comes from a stable model pair is automatically enriched over the stable homotopy category. To illustrate the general theory, we sketch a stable model pair construction of the stable homotopy category using a certain nonlinear comonad on the category of chain complexes of abelian groups (in this case, our fundamental spectral sequence becomes the Adams spectral sequence).
In principle, we could have finished the paper at this point, but it seemed useful to include two more things. In Sect. 11, we treat the case of algebraic triangulated categories; in the stable model setting, being algebraic corresponds to being klinear over a commutative ring k. We show that in such a linear situation, the stable model enhancement is very close to a DG enhancement, and we discuss the similarities and the differences between the two. We also show how our gluing machinery works in examples that come from algebraic geometry. Then in Sect. 12, we go back to Verdier localization theorem, and we prove its stable model counterpart. This could have been done much earlier in the paper, but since it uses Reedy model structures, we moved it to the end so as not to do the same thing twice. In the process of doing localization, we also construct an inductive completion of a stable model pair. In particular, we prove that the inductive completion of a rightproper model category has a natural model structure; this seems to be a new result.
One thing we do not seriously study in this paper is enhancement for triangulated functors. In Sect. 3.1, we give a short description of a class of functors that is sufficient for the purposes of gluing, and leave it at that. This certainly does not produce all functors, and this does not allow one to define a triangulated category of enhanced functors between two enhanced triangulated categories. Both in the \(\infty \)categorical and in the DG formalism, such a category is defined, and both definitions are highly nontrivial (in the DG setting, this is one of the main results of [44]). For some applications such as Hochschild homology, this is indispensable, and we do not know whether it is possible to do it using stable model pairs.
However, as far as gluing is concerned, our approach and the DG or \(\infty \)categorical approached seem to be of about equal strength. Both depend on a number of auxiliary choices. In the DG or \(\infty \)setting, the objects one deals with are exactly the objects in the homotopy category, and the choices occur at the level of morphisms: one has to fix a whole complex or simplicial set of maps, with the compositions or partial compositions. In our setting, one has to add more objects to form the ambient model category \(\mathcal {C}'\). However, it is just a category, its sets of morphisms are just sets, and the underlying homotopy types are produced automatically and canonically by the formalism. Whichever set of choices is less unpleasant depends on the specific problem at hand.
2 Stable model pairs
2.1 Homotopy categories
Following [2], we start with the notion of a relative category.
Definition 2.1
A relative category \(\langle \mathcal {C},W \rangle \) is a category \(\mathcal {C}\) equipped with a class of morphisms W that contains all identity maps. A functor between relative categories \(\langle \mathcal {C},W \rangle \), \(\mathcal {C}',W' \rangle \) is a functor \(F:\mathcal {C}\rightarrow \mathcal {C}'\) such that \(F(w) \in W'\) for any \(w \in W\).
Definition 2.2
A class of morphisms W in a category \(\mathcal {C}\) is closed under retracts if for any \(w:c \rightarrow c'\) in W and morphisms \(a:c \rightarrow c_0\), \(b:c_0 \rightarrow c\), \(a':c' \rightarrow c_0'\), \(b':c_0' \rightarrow c'\) such that \(a \circ b = \mathsf{id}\), \(a' \circ b' = \mathsf{id}\), the morhism \(a' \circ w \circ b:c_0 \rightarrow c_0'\) is in W. A class W is saturated if it is closed under retracts, and for any composable pair of morphisms f, g in \(\mathcal {C}\), if two out the three morphisms f, g, \(f \circ g\) are in W, then the third is also in W. A relative category \(\langle \mathcal {C},W \rangle \) is saturated if so is W.
Example 2.3
For any category \(\mathcal {C}\), denote by \({{\text {Iso}}}\) the class of all isomorphisms in \(\mathcal {C}\). Then \(\langle \mathcal {C},{{\text {Iso}}}\rangle \) is a saturated relative category. Any functor \(F:\mathcal {C}\rightarrow \mathcal {C}'\) is a functor \(F:\langle \mathcal {C},{{\text {Iso}}}\rangle \rightarrow \langle \mathcal {C}',{{\text {Iso}}}\rangle \).
We note that if a relative category \(\langle \mathcal {C},W \rangle \) is saturated, then automatically \({{\text {Iso}}}\subset W\). For any relative category \(\langle \mathcal {C},W \rangle \), we denote by \(\widehat{W}\) the minimal saturated class of maps in \(\mathcal {C}\) that contains W, and we call \(\langle \mathcal {C},\widehat{W} \rangle \) the saturation of \(\langle \mathcal {C},W \rangle \).
Definition 2.4
A relative category \(\langle \mathcal {C},W \rangle \) is localizable if there exists a category \({\text {Ho}}(\mathcal {C},W)\) and a functor \(h:\langle \mathcal {C},W \rangle \rightarrow \langle {\text {Ho}}(\mathcal {C}),{{\text {Iso}}}\rangle \) such that for any other category \(\mathcal {E}\), a functor \(F:\langle \mathcal {C},W \rangle \rightarrow \langle \mathcal {E},{{\text {Iso}}}\rangle \) factors as \(F = F' \circ h\) for some \(F':{\text {Ho}}(\mathcal {C},W) \rightarrow \mathcal {E}\), and this factorization is unique up to a unique isomorphism. In this case, \({\text {Ho}}(\mathcal {C},W)\) is called the homotopy category of \(\langle \mathcal {C},W \rangle \), and h is the localization functor.
For any relative category \(\langle \mathcal {C},W \rangle \), we have the opposite relative category \(\langle \mathcal {C}^o,W^o \rangle \), where \(W^o\) consists of maps \(w \in W\) considered as maps in \(\mathcal {C}^o\). The category \(\langle \mathcal {C},W \rangle \) is localizable if and only if so is \(\langle \mathcal {C}^o,W^o \rangle \), and in this case, we have \({\text {Ho}}(\mathcal {C},W)^o \cong {\text {Ho}}(\mathcal {C}^o,W^o)\).
Definition 2.5
The homotopy limit \(\mathsf{holim}_I F\) of a functor \(F:I \rightarrow \mathcal {C}\) is an object \(c \in {\text {Ho}}(\mathcal {C},W)\) equipped with a map \(f:{\text {Ho}}(\tau )(c) \rightarrow h(F)\) such that for any object \(c' \in {\text {Ho}}(\mathcal {C},W)\), any map \(f':{\text {Ho}}(\tau )(c') \rightarrow h(F)\) uniquely factors as \(f' = f \circ {\text {Ho}}(\tau )(g)\) for some map \(g:c' \rightarrow c\). The homotopy colimit \(\mathsf{hocolim}_I F\) is the homotopy limit of \(F^o:I^o \rightarrow \mathcal {C}^o\).
2.2 Model structures
Our notion of a model category is that of [39] (called “closed model category” there). To fix terminology, here is the definition.
Definition 2.6
 (i)
\(\mathcal {C}\) has finite limits and colimits,
 (ii)
C and F are closed under retracts and compositions,
 (iii)
C has a leftlifting property with respect to \(F \cap W\), and \(C \cap W\) has a leftlifting property with respect to F, in the sense of [39], and
 (iv)
every morphism g in \(\mathcal {C}\) decomposes as \(g = f_1 \circ c_1 = f_1 \circ c_2\), \(f_1 \in F\), \(f_2 \in F \cap W\), \(c_1 \in C \cap W\), \(c_2 \in C\).
Definition 2.7
Assume given a relative category \(\langle C,W \rangle \) equipped with a model structure \(\langle C,F \rangle \), and a small category I. A model structure \(\langle \widetilde{C},\widetilde{F} \rangle \) on \(\langle \mathcal {C}^I,W^I \rangle \) is injective if \(\widetilde{C} = C^I\) and projective if \(\widetilde{F}=F^I\).
We assume known that a model structure on a relative category \(\langle \mathcal {C},W \rangle \) is completely defined by either of the two classes C, F (F consists of morphisms that have the right lifting property with respect to \(C \cap W\), and C consists of morphisms that have the left lifting property with respect to \(F \cap W\)). Therefore the injective and projective model structures on \(\mathcal {C}^I\) are unique, if they exist. In general, they do not. One situation where they do exist is when I is the category \(\mathsf{V}\) used in the definition of homotopy cartesian and cocartesian squares. For any relative category \(\langle \mathcal {C},W \rangle \) equipped with a model structure, the relative category \(\langle \mathcal {C}^\mathsf{V},W^\mathsf{V}\rangle \) has the injective model structure, and \(\mathsf{holim}_\mathsf{V}E\) exists for any \(E \in {\text {Ho}}(\mathcal {C}^\mathsf{V},W^\mathsf{V})\). Dually, \(\langle \mathcal {C}^o,W^o \rangle \) has a model structure (with \(F^o\) as cofibrations and \(C^o\) as fibrations), and \(\mathsf{hocolim}_{\mathsf{V}^o} E\) exists for any \(E:\mathsf{V}^o \rightarrow \mathcal {C}\). To represent homotopy cartesian and cocartesian squares in \(\mathcal {C}\), it is convenient to introduce the following.
Definition 2.8
Then every homotopy cocartesian square in \(\mathcal {C}\) is weakly equivalent to a cofiber square, and conversely, every cofiber square is homotopy cocartesian. In fact, a cocartesian square (2.4) is homotopy cocartesian as soon as X is cofibrant, and either f or g is in C (for a proof, see e.g. [26] or Sect. 7.2). For fiber squares, the situation is dual.
For any integer \(n \ge 0\), denote by [n] the totally ordered set \(\{0,\dots ,n\}\) considered as a small category in the usual way. Then for any model structure on a relative category \(\langle \mathcal {C},W \rangle \), the category \(\langle \mathcal {C}^{[n]},W^{[n]} \rangle \) also has the injective and projective model structures. We will need the following simple lemma.
Lemma 2.9
Proof
We may assume that \(X_\cdot \) and \(X'_\cdot \) are fibrant and cofibrant for the projective model structure on \(\mathcal {C}^{[n]}\), so that \(f_\cdot ,f'_\cdot \in C\). Moreover, we may assume by induction that we already have the maps \(\widetilde{g}_i\) for \(i = 0,\dots ,n1\).
Choose a decomposition (2.3) for the object \(X_n \in \mathcal {C}\), and take an arbitrary map \(g:X_n' \rightarrow X_n\) such that \(h(g)=g_n\). Then \(h(g \circ f_{n1}') = h(f_{n1} \circ \widetilde{g}_{n1})\), so that there exists a map \(\widetilde{g}:X'_{n1} \rightarrow P(X_n)\) such that \(g \circ f_{n1}' = \pi _1 \circ \widetilde{g}\) and \(f_{n1} \circ \widetilde{g}_{n1} = \pi _2 \circ \widetilde{g}\). But \(\pi _1 \in F \cap W\), so by the lifting property, there exists a map \(q:X'_n \rightarrow P(X_n)\) such that \(\widetilde{g} = q \circ f'_{n1}\) and \(g = \pi _1 \circ q\). Take \(\widetilde{g}_n = \pi _2 \circ q\), and note that \(h(\widetilde{g}_n) = h(g) = g_n\), and \(\widetilde{g}_n \circ f'_{n1} = \pi _2 \circ q = f_{n1} \circ \widetilde{g}_{n1}\). \(\square \)
Finally, we make one general remark. If one takes a relative category \(\langle \mathcal {C},W \rangle \) as a primary object of study, then a model structure in the sense of Definition 2.6 is a choice. Making this choice ensures that \(\langle \mathcal {C},W \rangle \) is localizable and \({\text {Ho}}(\mathcal {C},W)\) is wellbehaved, but \({\text {Ho}}(\mathcal {C},W)\) itself does not depend on the choice made. Unfortunately, the choice is not always possible—for example, many relative categories needed in applications do not satisfy Definition 2.6 (i). To alleviate the problem, it is convenient to introduce the following.
Definition 2.10
A model embedding of a relative category \(\langle \mathcal {C},W \rangle \) is a fully faithful functor \(\langle \mathcal {C},W \rangle \rightarrow \langle \mathcal {C}',W' \rangle \) to a relative category \(\langle \mathcal {C}',W' \rangle \) equipped with a model structure such that for any weak equivalence \(w:X \rightarrow X'\), \(w \in W'\), we have \(X \in \mathcal {C}\subset \mathcal {C}'\) if and only if \(X' \in \mathcal {C}\subset \mathcal {C}'\).
Lemma 2.11
If a model category \(\langle \mathcal {C},W \rangle \) admits a model embedding \(\varepsilon :\langle \mathcal {C},W \rangle \rightarrow \langle \mathcal {C}',W \rangle \), then \(\langle \mathcal {C},W \rangle \) is localizable, and \({\text {Ho}}(\varepsilon ):{\text {Ho}}(\mathcal {C},W) \rightarrow {\text {Ho}}(\mathcal {C}',W')\) is fully faithful.
Proof
By virtue of Lemma 2.11, while \(\langle \mathcal {C},W \rangle \) itself does not necessarily have a model structure, it can still be studied by model category techniques. From the general point of view, this is only logical—if we need to choose a model structure anyway, why not choose also an ambient category that can carry it with some comfort.
2.3 Stability
If a localizable relative category \(\langle \mathcal {C},W \rangle \) is pointed, then \({\text {Ho}}(\mathcal {C},W)\) is also pointed, with the same object 0. We will say that \(X \in \mathcal {C}\) is acyclic if \(h(X) \cong 0\) (equivalently, \(X \rightarrow 0\) is in W, equivalently, \(0 \rightarrow X\) is in W).
Definition 2.12
 (i)
a square (2.4) with \(X,Y \in \mathcal {C}\) is homotopy cartesian if and only if it is homotopy cocartesian, and
 (ii)
in this case, \(X' \in \mathcal {C}\) if and only if \(Y' \in \mathcal {C}\).
Example 2.13
Assume given an abelian category \(\mathcal {C}\), let \(C_\cdot (\mathcal {C})\) be the category of chain complexes in \(\mathcal {C}\), and let W be the class of quasiisomorphisms. Then if \(\mathcal {C}\) has enough injectives, \(C_\cdot (\mathcal {C})\) has a model structure such that C consists of termwiseinjective maps, called the injective model structure, and if \(\mathcal {C}\) has enough projectives, it has the projective model structure with F formed by termwisesurjective maps. Both are stable. If we have an abelian subcategory \(\mathcal {C}_0 \subset \mathcal {C}\)—for example, coherent sheaves on a Noetherian scheme inside of quasicoherent schemes—and \(C_\cdot (\mathcal {C})\) has either the injective or the projective model structure, then \(\langle C_\cdot (\mathcal {C}_0),C_\cdot (\mathcal {C}) \rangle \) is a stable model pair.
Without the model embedding part, Definition 2.12 gives the very standard notion of a stable model category whose application to triangulated categories goes back at least to [26]. As it happens, most of the theory carries over to stable model pairs. We prove several simple results right away.
Lemma 2.14
For any stable model pair \(\langle \mathcal {C},W \rangle \subset \langle \mathcal {C}',W' \rangle \), the functors \(\Sigma \), \(\Omega \) of (2.7) send \({\text {Ho}}(\mathcal {C}) \subset {\text {Ho}}(\mathcal {C}')\) to itself and induce adjoint autoequivalences of \({\text {Ho}}(\mathcal {C})\).
Proof
Any object in \({\text {Ho}}(\mathcal {C})\) can be represented by a cofibrant object \(X \in \mathcal {C}\subset \mathcal {C}'\), and we can further choose a cofibration \(f:X \rightarrow Y\) with acyclic Y. Extend f to a cofiber square (2.4) in the sense of Definition 2.8, say with \(X = Y\). Then f represents S(h(X)) in \({\text {Ho}}(\mathcal {C}^{[2]},W^{[2]})\), and being cocartesian, (2.4) induces an isomorphism \(\Sigma (h(X)) \cong h(Y')\). Since \(X=Y\) is an acyclic object, and \(\mathcal {C}\subset \mathcal {C}'\) is a pointed model embedding, \(X=Y\) lies in \(\mathcal {C}\). Then by Definition 2.12 (ii), \(Y' \in \mathcal {C}\), and by Definition 2.12 (i), (2.4) is homotopy cartesian, thus induces an isomorphism \(h(X) \cong \Omega (h(Y'))\). Therefore \(\Sigma ({\text {Ho}}(C,W)) \subset {\text {Ho}}(\mathcal {C},W)\), and the adjunction map \(h(X) \rightarrow \Omega (\Sigma (h(X)))\) is an isomorphism for any \(X \in \mathcal {C}\). By the dual argument, \(\Omega ({\text {Ho}}(\mathcal {C},W)) \subset {\text {Ho}}(\mathcal {C},W)\), and the adjunction map \(\Sigma (\Omega (h(X))) \rightarrow h(X)\) is also an isomorphism for any \(X \in \mathcal {C}\). \(\square \)
Lemma 2.15
For any stable model pair \(\langle \mathcal {C},W \rangle \subset \langle \mathcal {C}',W' \rangle \), the category \({\text {Ho}}(\mathcal {C},W)\) has finite products and coproducts, and the map (2.5) is an isomorphism for any \(X,Y \in {\text {Ho}}(\mathcal {C},W)\).
Proof
For our final result, consider a cofibrant object \(X \in \mathcal {C}\), and form a cofiber square (2.4) with acyclic Y and \(X'\). Then it defines an isomorphism \(l:h(Y') \cong \Sigma (h(X))\). On the other hand, we can flip the square and consider it as a map from its top row to its bottom row. This also gives a canonical isomorphism \(r:h(Y') \cong \Sigma (h(X))\).
Lemma 2.16
For any stable model pair \(\langle \mathcal {C},W \rangle \subset \langle \mathcal {C}',W' \rangle \) and any cofiber square (2.4) in \(\mathcal {C}'\) with \(X \in \mathcal {C}\) and acyclic Y, \(X'\), the isomorphisms \(l,r:h(Y') \cong \Sigma (h(X))\) satisfy \(l+r=0\).
Proof
Corollary 2.17
The category \({\text {Ho}}(\mathcal {C})\) is additive.
Proof
By Lemma 2.15, \({\text {Ho}}(\mathcal {C})\) has finite sums and products that coincide, and spaces of maps in \({\text {Ho}}(\mathcal {C})\) are commutative monoids. It remains to check that they are abelian groups. Indeed, by Lemma 2.16, we have an automorphism \(\sigma :\Sigma \rightarrow \Sigma \) of the functor \(\Sigma \) such that \(\mathsf{Id}+ \sigma = 0\), so that any \(f:X \rightarrow Y\) has an inverse element \(f = \Omega (\sigma \circ \Sigma (f))\). \(\square \)
2.4 Triangulated structures
The main property of stable model pairs is that they produce triangulated categories. To state this, let us recall the precise definition.
Definition 2.18
 (TR1)
 (TR2)For any distinguished triangle (2.9), the triangle is distinguished.
 (TR3)For any distinguished triangle (2.9), another distinguished triangle and two maps \(x:X \rightarrow X'\), \(y:Y \rightarrow Y'\) such that \(y \circ f = f' \circ x\), there exists a map \(z:Z \rightarrow Z'\) such that \(z \circ g = g' \circ y\) and \(x[1] \circ q = q' \circ z\).
 (TR4)For any three objects \(X,Y,Z \in T\) and two maps \(f:X \rightarrow Y\), \(g:Y \rightarrow Z\), there exists a commutative diagram(2.10)
Formally, Definition 2.18 (TR4) is weaker than the octahedron axiom of [46]. However, as proved for instance in [19, Chapter IV, §1.4], the axioms (TR1)(TR3) imply that a distinguished triangle (2.9) is defined by \(f:X \rightarrow Y\) up to a (nonunique) isomorphism; this together with (TR4) is equivalent to the octahedron axiom in its usual form.
Remark 2.19
From time to time, people try to prove that (TR4) itself follows from the other axioms, or alternatively, that it can be dispensed with altogether without any noticable ill effects. One of the reasons for this is that historically, the first definition of a triangulated category in the literature appeared in [12], and the axiom (TR4) was not included. Let us mention that nevertheless, at least from the perspective of [46], this is a very strange way to look at things. Indeed, the whole point of introducing Definition 2.18 in [46] was that it has a nontrivial theorem attached to it—namely, the localization theorem that constructs the quotient \({\mathcal {D}}/{\mathcal {D}}_0\) of a triangulated category \({\mathcal {D}}\) by a triangulated subcategory \({\mathcal {D}}_0\) (we recall the precise formulation in Sect. 3.4). For this purpose, (TR4) is the most important axiom: it insures that the class of morphisms in \({\mathcal {D}}\) whose cone lies in \({\mathcal {D}}_0\) is saturated in the sense of Definition 2.2, and gives one control over the localization process.
Proposition 2.20
The category \({\text {Ho}}(\mathcal {C})\) with the shift \(X \mapsto X[1]\) and the class of distinguished triangles defined above is a triangulated category in the sense of Definition 2.18.
At least when \(\mathcal {C}=\mathcal {C}'\), this fact is extremely wellknown since at least [26], but short written proofs are not easy to come by. In [26] much more is proved, but extracting exactly the claim of Proposition 2.20 is a rather nontrivial exercise (one should also keep in mind the fact that in [26], triangulated categories are called “classical triangulated categories”, and a triangulated category in the sense [26, Definition 7.1.1] has a form of enhancement already hardcoded into it). More recently, a proof in the \(\infty \)categorical setting appears in [34], and a complete and detailed proof in the setting of derivators is worked out in [24]. However, just as [24, 26] proves much more than what we claim here, so the paper is rather long, and on the other hand, it still does not cover the case of model pairs. So we think that giving a short and selfcontained proof is useful. We emphasize that the argument is still the same, and claim no conceptual novelty whatsoever.
Proof of Proposition 2.20
Remark 2.21
Note that both stable model pairs and triangulated structures are selfdual notions, so that Proposition 2.20 actually provides two triangulated structures on \({\text {Ho}}(\mathcal {C},W)^o \cong {\text {Ho}}(\mathcal {C}^o,W^o)\). The triangles in the second structure are generated by diagrams (2.11) with fiber squares instead of cofiber squares. However, Definition 2.12 (i) immediately shows that the two structures coincide.
Lemma 2.22
Assume given a stable model pair \(\langle \mathcal {C},\mathcal {C}' \rangle \) and a full triangulated subcategory \({\mathcal {D}}_0 \subset {\mathcal {D}}= {\text {Ho}}(\mathcal {C})\), and let \(\mathcal {C}_0 \subset \mathcal {C}\) be the full subcategory spanned by objects X with \(h(X) \in {\mathcal {D}}_0\). Then \(\langle \mathcal {C}_0,\mathcal {C}' \rangle \) is a stable model pair.
Proof
By definition, \(\mathcal {C}_0 \subset \mathcal {C}'\) is a model embedding, and Definition 2.12 (i) follows from the corresponding property of \(\mathcal {C}\subset \mathcal {C}'\). For Definition 2.12 (ii), note that the fourth term in a homotopy cartesian or cocartesian square (2.4) can be expressed as a cone of a map between sums of shifts of the other three terms, and \({\mathcal {D}}\subset {\text {Ho}}(\mathcal {C})\) is by assumption stable under cones, shifts and finite sums. \(\square \)
3 Elementary gluing
3.1 Functors
The standard class of functors between model categories that induce natural derived functors on the homotopy level are the left and rightQuillen functors. However, for some applications this notion is too strong. Therefore let us introduce the following.
Definition 3.1
A functor \(\Phi :\langle \mathcal {C}_0,W_0 \rangle \rightarrow \langle \mathcal {C}_1,W_1 \rangle \) between two relative categories with model structures \(C_0,F_0\) and \(C_1,F_1\) is leftderivable if it sends morphisms in \(C_0\) to morphisms in \(C_1\) and morphisms in \(C_0 \cap W_0\) to morphisms in \(C_1 \cap W_1\). A functor \(\Phi \) is rightderivable if \(\Phi ^o\) is leftderivable.
The difference between leftderivable functors of Definition 3.1 and leftQuillen functors is exactness: we do not require our functors to preserve finite colimits. Still, as in [39], any leftderivable functor \(\Phi :\mathcal {C}_0 \rightarrow \mathcal {C}_1\) induces a natural leftderived functor \(L^\cdot \Phi :{\text {Ho}}(\mathcal {C}_0,W_0) \rightarrow {\text {Ho}}(\mathcal {C}_1,W_1)\) obtained by restricting \(\Phi \) to the full subcategory spanned by cofibrant objects in \(\mathcal {C}_0\). Analogously, a rightderivable functor \(\Phi \) induces a rightderived functor \(R^\cdot \Phi :{\text {Ho}}(\mathcal {C}_0,W_0) \rightarrow {\text {Ho}}(\mathcal {C}_1,W_1)\). In these terms, the Quillen Adjunction Theorem can be restated as follows: if \(\Phi ^\dagger \) is rightadjoint to \(\Phi \), and \(\Phi \) is leftderivable, then \(\Phi ^\dagger \) is rightderivable, and \(R^\cdot \Phi ^\dagger \) is rightadjoint to \(L^\cdot \Phi \). Another trivial but useful observation is the following: if a functor \(\Phi :\mathcal {C}_0 \rightarrow \mathcal {C}_1\) is both left and rightderivable, then it sends W to W, thus induces a functor \({\text {Ho}}(\Phi ):{\text {Ho}}(\mathcal {C}_0) \rightarrow {\text {Ho}}(\mathcal {C}_1)\), and we have \(L^\cdot \Phi \cong {\text {Ho}}(\Phi ) \cong R^\cdot \Phi \).
Definition 3.2
Assume given stable model pairs \(\langle \mathcal {C}_0',\mathcal {C}_0 \rangle \), \(\langle \mathcal {C}_1',\mathcal {C}_1 \rangle \). A left resp. rightderivable functor \(\Phi :\mathcal {C}_0 \rightarrow \mathcal {C}_1\) is stable if it is pointed, and \(L^\cdot \Phi \) resp. \(R^\cdot \Phi \) sends \({\text {Ho}}(\mathcal {C}_0') \subset {\text {Ho}}(\mathcal {C}_0)\) into \({\text {Ho}}(\mathcal {C}_1') \subset {\text {Ho}}(\mathcal {C}_1)\) and homotopy cartesian squares of Definition 2.12 (i) in \({\text {Ho}}(\mathcal {C}_0')\) to homotopy cartesian squares in \({\text {Ho}}(\mathcal {C}_1')\).
Remark 3.3
By Definition 2.12 (i), replacing homotopy cartesian squares in Definition 3.2 with homotopy cocartesian squares gives the same notion.
Remark 3.4
In the assumptions of Definition 3.2, a rightQuillen functor \(\Phi :\mathcal {C}_0 \rightarrow \mathcal {C}_1\) is obviously stable as soon as \(R^\cdot \Phi \) sends \({\text {Ho}}(\mathcal {C}_0')\) into \({\text {Ho}}(\mathcal {C}_1')\).
Example 3.5
Assume given a leftexact functor \(\Phi :\mathcal {C}\rightarrow \mathcal {E}\) between Grothendieck abelian categories. Then the induced functor \(\Phi _\cdot :C_\cdot (\mathcal {C}) \rightarrow C_\cdot (\mathcal {E})\) is rightQuillen, hence also rightderivable with respect to the injective model structures, and \(R^\cdot \Phi _\cdot \) is the usual derived functor of the functor \(\Phi \).
Example 3.6
Lemma 3.7
Proof
3.2 Simple gluing
Proposition 3.8
 (i)
Assume given a functor \(\Phi :\mathcal {C}_0 \rightarrow \mathcal {C}_1\) between two model categories that is rightderivable in the sense of Definition 3.1. Then the commacategory \(\mathsf{R}(\Phi )\) has a (unique) model structure such that \(\langle f_0,f_1 \rangle \) lies in W resp. C if and only if so do \(f_0\) and \(f_1\). Moreover, if \(\langle f_0,f_1 \rangle \) is in F with respect to this model structure, then so are \(f_0\) and \(f_1\).
 (ii)
Assume given stable model pairs \(\langle \mathcal {C}_0',\mathcal {C}_0 \rangle \), \(\langle \mathcal {C}_1',\mathcal {C}_1 \rangle \), assume that \(\Phi \) is stable in the sense of Definition 3.2, and let \(\mathsf{R}(\Phi )' \subset \mathsf{R}(\Phi )\) the full subcategory spanned by triples \(\langle c_0,c_1,\alpha \rangle \) with \(c_0 \in \mathcal {C}_0'\) and \(c_1 \in \mathcal {C}_1'\). Then \(\langle \mathsf{R}(\Phi )',\mathsf{R}(\Phi ) \rangle \) is a stable model pair.
In the context of stable model pairs, it is Proposition 3.8 (ii) that gives a solution to the elementary gluing problem mentioned in Sect. 1.2. We note that already the following fact is slightly counterintuitive.
Lemma 3.9
In the assumptions of Proposition 3.8 (i), the category \(\mathsf{R}(\Phi )\) has finite limits and colimits.
Proof
Proof of Proposition 3.8
Finally, for Definition 2.6 (iv), assume given a map \(X \rightarrow Y\) in \(\mathsf{R}(\Phi )\). To find either of the two necessary factorizations \(X \rightarrow Z \rightarrow Y\) for this map, construct first the corresponding factorization \(X_0 \rightarrow Z_0 \rightarrow Y_0\) in \(\mathcal {C}_0\), and then factorize the induced map \(X_1 \rightarrow \Phi (Z_0) \times _{\Phi (Y_0)} Y_1\) in \(\mathcal {C}_1\).
3.3 Twosided gluing
Since the notions of a model category and a stable model pair are selfdual, Proposition 3.8 immediately implies an analogous statement for left commacategories and leftderivable functors. However, it turns out that there is more: one can combine Proposition 3.8 and its dual into a single statement. Namely, assume given two categories \(\mathcal {C}_0\), \(\mathcal {C}_1\), two functors \(L,M:\mathcal {C}_0 \rightarrow \mathcal {C}_1\), and a morphism \(q:L \rightarrow M\).
Definition 3.10
The twosided commacategory \(\mathsf{G}(L,M,q)\) is the category of quadruples \(\langle c_0,c_1,l,m \rangle \) of objects \(c_0 \in \mathcal {C}_0\), \(c_1 \in \mathcal {C}_1\) equipped with morphisms \(l:L(c_0) \rightarrow c_1\), \(m:c_1 \rightarrow M(c_0)\) such that \(m \circ l = q\).
Example 3.11
For a reallife example of a twosided commacategory, consider the standard description of the category of perverse sheaves on a formal disc D, as in e.g. [3]. As \(\mathcal {C}_0\), we take the category of local systems on the punctured disc \(\overline{D}\), \(\mathcal {C}_1\) is the category of sheaves on the central point, \(L=M\) is the nearby cycles functor \(\Psi \), and q is identity minus the monodromy operator. Then a perverse sheaf E on the disc corresponds to the quadruple \(\langle E_{\overline{D}},\Phi (E), \mathsf{can}, \mathsf{var}\rangle \), where \(\Phi \) is the vanishing cycles functor, and \(\mathsf{can}\), \(\mathsf{var}\) are the usual functorial maps.
Proposition 3.12
 (i)
Assume that \(\mathcal {C}_0\), \(\mathcal {C}_1\) are model categories, L is leftderivable, and M is rightderivable, and equip \(\mathsf{L}(L)\), \(\mathsf{R}(M)\) with the model structures of Proposition 3.8. Say that a map f in \(\mathsf{G}(L,M,q)\) is in C resp. F if so is \(\lambda ^*(f)\) resp. \(\mu ^*(f)\), and say that f is in W if \(\lambda ^*(f)\) is in W (or equivalently, \(\mu ^*(f)\) is in W). Then C, F, W is a model structure on \(\mathsf{G}(L,M,q)\).
 (ii)
Assume given stable model pairs \(\langle \mathcal {C}_0,\mathcal {C}_0'\rangle \), \(\langle \mathcal {C}_1,\mathcal {C}_1' \rangle \), assume that the functors L and M are stable in the sense of Definition 3.2, and let \(\mathsf{G}(L,M,q)' \subset \mathsf{G}(L,M,q)\) be the full subcategory spanned by quadruples \(\langle c_0,c_1,l,m \rangle \) with \(c_0 \in \mathcal {C}_0'\), \(c_1 \in \mathcal {C}_1'\). Then \(\langle \mathsf{G}(L,M,q)', \mathsf{G}(L,M,q) \rangle \) is a stable model pair.
Proof
For (i), note that a map in \(\mathsf{G}(L,M,q)\) is a weak equivalence iff it is a weak equivalence componentwise, so that \(\mathsf{G}(L,M,q)\) is trivially saturated, and moreover, Definition 2.6 (ii) tautologically follows from the corresponding property of the model structures on \(\mathsf{L}(L)\) and \(\mathsf{R}(M)\).
For (ii), observe that again as in Proposition 3.8 it suffices to prove that a square (2.4) is homotopy cartesian resp. cocartesian if and only if so are the squares (3.5). In the cartesian case, note that \(\mu ^*\) sends weak equivalence to weak equivalences, fibrations to fibrations, and cartesian squares to cartesian squares, so that (2.4) is homotopy cartesian iff it becomes homotopy cartesian after applying the functor \(\mu ^*\). The claim then immediately follows from the corresponding claim in the proof of Proposition 3.8. In the cocartesian case, use the functor \(\lambda ^*\). \(\square \)
3.4 Semiorthogonal decompositions
To understand better what are the implications of Proposition 3.12 (ii) on the level of homotopy categories, let us briefly recall the Verdier localization formalism and the notion of a semiorthogonal decomposition.
A full triangulated subcategory \({\mathcal {D}}_0 \subset {\mathcal {D}}\) in a triangulated category \({\mathcal {D}}\) is called thick, or sometimes saturated, if it is closed under retracts ([46] has a formally different notion of an “epaisse” subcategory that turns out to be equivalent).
Theorem 3.13
 (i)
The class W of maps f in \({\mathcal {D}}\) whose cone \({\text {Cone}}(f)\) lies in \({\mathcal {D}}_0\) is saturated in the sense of Definition 2.2.
 (ii)
Assume that the relative category \(\langle {\mathcal {D}},W \rangle \) is localizable in the sense of Definition 2.4. Then the homotopy category \({\text {Ho}}({\mathcal {D}},W)\) is triangulated, the quotient functor \(h:{\mathcal {D}}\rightarrow {\text {Ho}}({\mathcal {D}},W)\) is a triangulated functor, and for any object \(E \in {\mathcal {D}}\), \(h(E)=0\) if and only if E lies in \({\mathcal {D}}_0 \subset {\mathcal {D}}\). \(\square \)
Definition 3.14
A full triangulated subcategory \({\mathcal {D}}_0 \subset {\mathcal {D}}\) in a triangulated category \({\mathcal {D}}\) is rightlocalizing if for any \(E \in {\mathcal {D}}\), there exists a filtered small category I and a cofinal functor \(I \rightarrow W(E)\), and leftlocalizing if \({\mathcal {D}}^o_0 \subset {\mathcal {D}}^o\) is rightlocalizing.
If \({\mathcal {D}}_0 \subset {\mathcal {D}}\) is rightlocalizing, then (3.8) gives a welldefined set, so that \(\langle {\mathcal {D}},W \rangle \) is localizable, and the quotient \({\mathcal {D}}/{\mathcal {D}}_0\) exists. Dually, if \({\mathcal {D}}_0 \subset {\mathcal {D}}\) is leftlocalizing, the quotient \({\mathcal {D}}/{\mathcal {D}}_0 = ({\mathcal {D}}^o/{\mathcal {D}}_0^o)^o\) also exists. A small subcategory \({\mathcal {D}}_0 \subset {\mathcal {D}}\) is always both right and leftlocalizing: up to an isomorphism, an object \(w:E' \rightarrow E\) in W(E) is given by an object \({\text {Cone}}(w) \in {\mathcal {D}}_0\) and a morphism \(E \rightarrow {\text {Cone}}(w)\), so that W(E) itself is essentially small. Another example of a localizing triangulated subcategory appears in the situation of a semiorthogonal decomposition introduced in [6].
Definition 3.15
 (i)
\({\text {Hom}}(A,B) = 0\) for any \(A \in {\mathcal {D}}_0 \subset {\mathcal {D}}\), \(B \in {\mathcal {D}}_1 \subset {\mathcal {D}}\), and
 (ii)any object \(E \in {\mathcal {D}}\) fits into a distinguished triangle(3.9)
One shows that in the situation of Definition 3.15, \(E_0\) and \(E_1\) in (3.9) are functorial in E, and sending E to \(E_0\) resp. \(E_1\) gives functors \(R:{\mathcal {D}}\rightarrow {\mathcal {D}}_0\), \(L:{\mathcal {D}}\rightarrow {\mathcal {D}}_1\) right resp. leftadjoint to the embeddings, with r and l of (3.9) being the adjunction map. Both \({\mathcal {D}}_0\) and \({\mathcal {D}}_1\) are automatically thick, \({\mathcal {D}}_0 \subset {\mathcal {D}}\) is rightlocalizing, \({\mathcal {D}}_1 \subset {\mathcal {D}}\) is leftlocalizing, and L, R induce equivalences \({\mathcal {D}}_1 \cong {\mathcal {D}}/{\mathcal {D}}_0\), \({\mathcal {D}}_0 \cong {\mathcal {D}}/{\mathcal {D}}_1\).
Conversely, a full triangulated subcategory \({\mathcal {D}}_0 \subset {\mathcal {D}}_1\) is called left resp. rightadmissible if the embedding functor admits a left resp. rightadjoint, and simply admissible if it is left and rightadmissible. Then any rightadmissible full triangulated subcategory \({\mathcal {D}}_0 \subset {\mathcal {D}}\) is automatically thick and rightlocalizing, the quotient functor \({\mathcal {D}}\rightarrow {\mathcal {D}}_1 = {\mathcal {D}}/{\mathcal {D}}_0\) has a fully faithful rightadjoint \({\mathcal {D}}_1 \rightarrow {\mathcal {D}}\), and \(\langle {\mathcal {D}}_0,{\mathcal {D}}_1 \rangle \) is a semiorthogonal decomposition of the category \({\mathcal {D}}\). Dually, for a leftadmissible subcategory \({\mathcal {D}}_0 \subset {\mathcal {D}}\), we have a leftadjoint \({\mathcal {D}}_1 = {\mathcal {D}}/{\mathcal {D}}_0 \rightarrow {\mathcal {D}}\) to the quotient functor, and a semiorthogonal decomposition \(\langle {\mathcal {D}}_1,{\mathcal {D}}_0 \rangle \).
For an admissible subcategory \({\mathcal {D}}_0 \subset {\mathcal {D}}\), we can actually compose the embedding \({\mathcal {D}}/{\mathcal {D}}_0 \rightarrow {\mathcal {D}}\) rightadjoint to the quotient functor with the leftadjoint \({\mathcal {D}}\rightarrow {\mathcal {D}}_0\) to the embedding, and obtain a triangulated functor \(F:{\mathcal {D}}/{\mathcal {D}}_0 \rightarrow {\mathcal {D}}_0\). This is known as the left gluing functor of the semiorthogonal decomposition \(\langle {\mathcal {D}}_0,{\mathcal {D}}/{\mathcal {D}}_0 \rangle \). Alternatively, we can take the leftadjoint \({\mathcal {D}}/{\mathcal {D}}_0 \rightarrow {\mathcal {D}}\) and the rightadjoint \({\mathcal {D}}\rightarrow {\mathcal {D}}_0\); this gives the right gluing functor \(F':{\mathcal {D}}/{\mathcal {D}}_0 \rightarrow {\mathcal {D}}_0\).
Now, doing a version of Theorem 3.13 for stable model pairs is rather delicate, and we will do it in the end of the paper, in Sect. 12. However, in the situation of a semiorthogonal decomposition, things are straightforward. Namely, if we have a stable model pair \(\langle \mathcal {C},\mathcal {C}' \rangle \) and a semiorthogonal decomposition \(\langle {\mathcal {D}}_0,{\mathcal {D}}_1 \rangle \) of the triangulated category \({\mathcal {D}}= {\text {Ho}}(\mathcal {C})\), then the preimages \(\mathcal {C}_0,\mathcal {C}_1 \subset \mathcal {C}\) of the subcategories \({\mathcal {D}}_0,{\mathcal {D}}_1 \subset {\mathcal {D}}\) under the localization functor \(h:\mathcal {C}\rightarrow {\mathcal {D}}\) trivially fit into stable model pairs \(\langle \mathcal {C}_0,\mathcal {C}' \rangle \), \(\langle \mathcal {C}_1,\mathcal {C}' \rangle \).
Let us now show that in a sense, Proposition 3.12 (ii) reverses the construction and produces a triangulated category starting from gluing data for a semiorthogonal decomposition.
Lemma 3.16
Proof
The fact that U and V are both left and rightderivable follows immediately from the definition of the model structure on \(\mathsf{G}(L',M',q')\). Then by adjunction, \(U_\dagger \) and \(V_\dagger \) resp. \(U^\dagger \) and \(V^\dagger \) are left resp. rightderivable. Moreover, all these functors are in fact Quillen, so that they are all automatically stable, and we obtain the required adjunctions on the homotopy category level by the Quillen Adjunction Theorem. The functor V is also obviously fully faithful, and we have \(R^\cdot V^\dagger \circ R^\cdot V \cong R^\cdot (V^\dagger \circ V) \cong \mathsf{id}\), so that \(L^\cdot V \cong R^\cdot V\) is still fully faithful. The distinguished triangles (3.12) are induced by the squares (3.11). Finally, the functors \(U_\dagger \) and \(U^\dagger \) are also obviously fully faithful, and since U is both left and rightderivable, their derived functors are fully faithful as well. \(\square \)
4 Families of categories
4.1 Grothendieck construction
Definition 4.1
A morphism \(f:c' \rightarrow c\) in a category \(\mathcal {C}\) is vertical with respect to a functor \(\pi :\mathcal {C}\rightarrow I\) if \(\pi (f)=\mathsf{id}\), and cartesian with respect to \(\pi \) if any \(f':c'' \rightarrow c\) such that \(\pi (f) = \pi (f')\) uniquely factorizes as \(f' = f \circ f_0\) with vertical \(f_0\). A cartesian lifting of a morphism \(f:i' \rightarrow i\) in I is a morphism \(f'\) in \(\mathcal {C}\) cartesian with respect to \(\pi \) and such that \(\pi (f')=f\). A functor \(\pi :\mathcal {C}\rightarrow I\) is a prefibration if for any \(c \in \mathcal {C}\), any morphism \(f:i' \rightarrow i=\pi (c)\) in \(\mathcal {E}\) admits a cartesian lifting \(f':c' \rightarrow c\). A prefibration is a fibration if the composition of cartesian morphisms is cartesian. A morphism is cocartesian if it is cartesian as a morphism in the opposite category \(\mathcal {C}^o\) with respect to the opposite functor \(\pi ^o:\mathcal {C}^o \rightarrow I^o\), and a functor \(\pi \) is a precofibration resp. cofibration if the opposite functor \(\pi ^o\) is a prefibration resp. fibration. A functor \(\pi \) is a bifibration if it is both a fibration and a cofibration.
Remark 4.2
In recent retellings of [21] such as [47], it has become usual to strengthen the definition of a cartesian map so that every prefibration is automatically a fibration. However, for our purposes, prefibrations are just as important.
Remark 4.3
 (i)
In Sects. 4, 5 and 6, model structures do not appear at all. In these Sections, “fibration” resp. “cofibration” means fibration resp. cofibration in the sense of Definition 4.1.
 (ii)
From Sect. 7 onward, “fibration” or “cofibration” without qualifiers is always used in the model category sense, and fibrations resp. cofibrations of Definition 4.1 are Grothendieck fibrations resp. Grothendieck cofibrations.
Example 4.4
For any functor \(\Phi :\mathcal {C}_0 \rightarrow \mathcal {C}_1\) between two categories \(\mathcal {C}_0\), \(\mathcal {C}_1\), the projection \(\tau :\mathsf{R}(\Phi ) \rightarrow \mathcal {C}_1\) of (3.2) is a fibration, and dually, the projection \(\tau :\mathsf{L}(\Phi ) \rightarrow \mathcal {C}_1\) is a cofibration. A map f is cartesian resp. cocartesian with respect to \(\tau \) iff \(\sigma (f)\) is invertible.
Definition 4.5
 (i)
a category F(i) for any object \(i \in I\),
 (ii)
a functor \(F(f):F(i) \rightarrow F(i')\) for any morphism \(f:i \rightarrow i'\) in I,
 (iii)
a morphism \(\alpha (i):F(\mathsf{id}_i) \rightarrow \mathsf{id}_{F(i)}\) for any identity map \(\mathsf{id}_i:i \rightarrow i\), and
 (iv)
a morphism \(\alpha (f_1,f_2):F(f_1 \circ f_2) \rightarrow F(f_1) \circ F(f_2)\) for any composable pair of maps \(f_1\), \(f_2\),
Then in this terminology, a prefibration I defines a normalized contravariant lax functor from I to \({\text {Cat}}\).
A prefibration is a fibration if and only if all the maps (4.2) are invertible. In this case, the corresponding lax functor reduces to what Grothendieck called a contravariant pseudofunctor from I to \({\text {Cat}}\). Conversely, for any such pseudofunctor \(F:I^o \rightarrow {\text {Cat}}\), one can consider the category \(\mathcal {C}\) of pairs \(\langle i,c \rangle \), \(i \in I\), \(c \in F(i)\), with maps from \(\langle i,c \rangle \) to \(\langle i',c' \rangle \) given by a map \(f:i \rightarrow i'\) and a map \(c \rightarrow F(f)(c')\), and composition defined using the morphisms \(\alpha (f,f')\). The two constructions are inverse to each other (nowadays this is known as the Grothendieck construction). The same inverse construction identifies prefibrations and normalized lax functors.
Remark 4.6
Strictly speaking, transition functors \(f^*\) of a prefibration are welldefined only up to a unique isomorphism, not uniquely. To convert a prefibration into a lax functor, one has to actually fix the functors \(f^*\) (in Grothendieck’s terminology, this is called choosing a cleavage of a prefibration). However, lax functors obtained from different cleavages are canonically isomorphic.
Example 4.7
Let \(\Delta \) be the category of finite ordinals \([n] = \{0,\dots ,n\}\), \(n \ge 0\) and orderpreserving maps between them. Note that we can treat an ordinal as a small category in the usual way, and then this is firstly, consistent with our earlier notation, and secondly, gives an embedding \(\Delta \subset {\text {Cat}}\). The Grothendieck construction then associates a cofibration \(\nu :\Delta ^\flat \rightarrow \Delta \) to this embedding. Objects of the category \(\Delta ^\flat \) are pairs \(\langle [n],l \rangle \), \([n] \in \Delta \), \(l \in [n]\), and maps from \(\langle [n],l \rangle \) to \(\langle [n'],l' \rangle \) given by maps \(f:[n] \rightarrow [n']\) such that \(f(l) \le l'\). The functor \(\nu \) is the forgetful functor sending \(\langle [n],l \rangle \) to [n].
Alternatively, one can send \([n] \in \Delta \) to the opposite category \([n]^o\). This gives the category \(\Delta _\flat \) of pairs \(\langle [n],l \rangle \) and maps \(f:[n] \rightarrow [n']\) such that \(f(l) \ge l'\), again with the forgetful functor \(\nu :\Delta _\flat \rightarrow \Delta \).
Remark 4.8
The category \(\Delta \) is embedded into the larger category of all finite nonempty totally ordered sets, and although the embedding is an equivalence, it is prudent to distinguish between the two (as e.g. in [13]). We will sometimes fail to do so and assume that, by abuse of notation, any abstract finite nonempty ordinal defines an object in \(\Delta \). Doing this consistently amounts to fixing once and for all an equivalence inverse to the embedding above, and it is certainly possible to do this in any reasonable foundational context.
Example 4.9
A useful source of prefibrations is the following construction. Assume given a prefibration \(\mathcal {C}\rightarrow I\) and a full subcategory \(\mathcal {C}' \rightarrow I\). Moreover, assume that either for any object \(i \in I\), the embedding \(\mathcal {C}'_i \rightarrow \mathcal {C}_i\) admits a rightadjoint functor r(i), or for any morphism \(f:i \rightarrow i'\), the transition functor \(f^*:\mathcal {C}_{i'} \rightarrow \mathcal {C}_i\) sends \(\mathcal {C}'_{i'} \subset \mathcal {C}_{i'}\) into \(\mathcal {C}'_i \subset \mathcal {C}_i\). Then the induced functor \(\mathcal {C}' \rightarrow I\) is also a prefibration. In the second case, its transition functors are simply induced by the transition functors \(f^*\) of \(\mathcal {C}\rightarrow I\), and in the first case, they are given by \(r(i) \circ f^*\) for any \(f:i \rightarrow i'\).
4.2 Functors and sections
Assume given categories \(\mathcal {C}\), \(\mathcal {C}'\) and functors \(\pi :\mathcal {C}\rightarrow I\), \(\pi ':\mathcal {C}' \rightarrow I\) to a category I.
Definition 4.10
 (i)
A functor from \(\mathcal {C}\) to \(\mathcal {C}'\) over I is a functor \(F:\mathcal {C}\rightarrow \mathcal {C}'\) equipped with an isomorphism \(\pi ' \circ F \cong \pi \).
 (ii)
Assume that \(\pi \), \(\pi '\) are prefibrations. A functor F from \(\mathcal {C}\) to \(\mathcal {C}'\) is cartesian along a morphism \(f:i \rightarrow i'\) in I if it sends cartesian liftings of f to cartesian liftings of f. A functor F is cartesian if it sends all cartesian maps to cartesian maps.
Lemma 4.11
 (i)
The composition functor \(\gamma \circ \pi :\mathcal {C}\rightarrow I\) is a prefibration, and \(\pi \) is cartesian over I.
 (ii)
The map (4.2) for the prefibration \(\pi \) is an isomorphism as soon as \(f_1\) is cartesian and \(f_2\) is vertical over I.
Proof
It immediately follows from the definition that any map f in \(\mathcal {C}\) cartesian over I must be also cartesian over \(I'\). Therefore (i) is equivalent to saying that any map f in \(\mathcal {C}\) cartesian over \(I'\) and such that \(\pi (f)\) is cartesian over I is itself cartesian over I. Let \(f:c' \rightarrow c\) be such a map. Then since \(\pi \) is a prefibration, any other map \(f':c'' \rightarrow c\) in \(\mathcal {C}\) must factor uniquely as \(f' = f'' \circ v\) with \(f''\) cartesian and v vertical over \(I'\), and if \(\gamma (\pi (f)) = \gamma (\pi (f'))\), then we must also have \(\pi (f') = \pi (f) \circ v'\), where \(v':\pi (c'') \rightarrow \pi (c')\) is vertical over I. Now, if we denote by \(v'':v^*c' \rightarrow c'\) the cartesian lifting of the map \(v'\), then the cartesian property of f over I is equivalent to saying that \(f'\) factors uniquely through \(f \circ v''\) by means of a map vertical over \(I'\), so that \(f \circ v''\) has the same universal property as \(f''\). Then we must have \(f \circ v'' = f''\), and this is equivalent to (ii). \(\square \)
Lemma 4.12
 (i)
The composition functor \(\gamma \circ \pi :\mathcal {C}\rightarrow I\) is a precofibration, and \(\pi \) is cocartesian over I.
 (ii)
The map (4.2) for the prefibration \(\pi \) is an isomorphism as soon as \(f_2\) is cocartesian and \(f_1\) is vertical over I, and for any map f in \(I'\) cocartesian over I, the transition functor \(f^*\) admits a leftadjoint \(f_!\).
Proof
By definition, (i) is equivalent to saying that for any \(c \in \mathcal {C}\) and map \(f:\pi (c) \rightarrow i'\) cocartesian over I, there exists a map \(f':c \rightarrow c'\) that lifts f (that is, \(\pi (f')=f\)), and is cocartesian both over \(I'\) and over I. Existence of lifting maps \(f'\) cocartesian over \(I'\) is equivalent to saying that \(f^*\) has a leftadjoint \(f_!\). As in Lemma 4.2, the condition on the maps (4.2) is then equivalent to saying that these maps \(f'\) are also cocartesian over I. \(\square \)
Example 4.13
Let \(I=[1]^o\). By definition, a prefibration \(\mathcal {C}\) over I consists of two categories \(\mathcal {C}_0\), \(\mathcal {C}_1\), and a transition functor \(\Phi :\mathcal {C}_0 \rightarrow \mathcal {C}_1\). The category \({\text {Sec}}([1]^o,\mathcal {C})\) is then precisely the right commacategory \(\mathsf{R}(\Phi )\), and the evaluation functors \(\mathsf{ev}_0\) resp. \(\mathsf{ev}_1\) send \(\langle c_0,c_1,\alpha \rangle \) to \(c_0\) resp. \(c_1\).
Example 4.14
In the situation of Example 4.13, assume that \(\mathcal {C}_0\) and \(\mathcal {C}_1\) are equipped with prefibrations \(\mathcal {C}_0,\mathcal {C}_1 \rightarrow I\), and \(\Phi \) is a functor over I. Then \(\mathcal {C}\rightarrow I \times [1]^o\) is also a prefibration, and we have \({\text {Sec}}(I \times [1],\mathcal {C}) \cong \mathsf{R}({\text {Sec}}(I,\Phi ))\), where \({\text {Sec}}(I,\Phi ):{\text {Sec}}(I,\mathcal {C}_0) \rightarrow {\text {Sec}}(I,\mathcal {C}_1)\) is the functor induced by \(\Phi \).
A morphism \(\varphi :\sigma \rightarrow \sigma '\) between two sections \(\sigma ,\sigma ' \in {\text {Sec}}(I,\mathcal {C})\) of prefibration \(\pi :\mathcal {C}\rightarrow I\) is by definition given by a collection of maps \(\varphi (i):\sigma (i) \rightarrow \sigma '(i)\), \(i \in I\) that commute with the maps (4.6) in the sense that for \(f:i \rightarrow i'\) in I, we have \(f^*(\varphi (i')) \circ \sigma (f) = \sigma '(f) \circ \varphi (i)\). The same data can be packaged in yet another way that we will use.
Definition 4.15
4.3 Pullbacks and adjunction
Lemma 4.16
Proof
For the second result, assume that we have two functors \(\gamma _0,\gamma _1:I' \rightarrow I\) and a prefibration \(\mathcal {C}\rightarrow I\). Observe that giving map \(a:\gamma _0 \rightarrow \gamma _1\) is equivalent to giving a functor \(\widetilde{a}:I' \times [1] \rightarrow I\) that restricts to \(\gamma _0\) resp. \(\gamma _1\) on \(I' \times 0 \subset I' \times [1]\) resp. \(I' \times 1 \subset I' \times [1]\).
Definition 4.17
The map a is compatible with the prefibration \(\mathcal {C}\) if for any morphism \(f:i \rightarrow i'\) in \(I'\), the natural map \(\gamma _0(f)^* \circ a(i')^* \rightarrow (a(i') \circ \gamma _0(f))^*\) is an isomorphism, and strictly compatible with \(\mathcal {C}\) if the maps \(a(i)^* \circ \gamma _1(f)^* \rightarrow (\gamma _1(f) \circ a(i))^*\) are isomorphisms as well.
Lemma 4.18
Proof
4.4 Transpose fibrations
4.5 Categorical gluing
Extrapolating from Example 4.13, one could expect that for a general normalized covariant lax functor from some small category I to \({\text {Cat}}\), the appropriate gluing is given by the category of sections \({\text {Sec}}(I,\mathcal {C})\) of the corresponding precofibration \(\mathcal {C}\rightarrow I\). However, this is not what happens. As a sign of this, we observe that Example 4.13 cannot be iterated: the maps (4.1) and (4.2) go in the opposite directions. Thus a collection of categories \(\mathcal {C}_i\), \(i=0,1,2\) and functors \(\Phi _{ij}:\mathcal {C}_i \rightarrow C_j\), \(0 \le i <j \le 2\) related by a map (4.1) form a precofibration over the category \([2] = \{0,1,2\}\), not a prefibration, and the category of sections of this precobiration has nothing to do with the iterated commacategory \(\mathsf{R}(\Phi _\cdot )\).
To find the correct notion of a good gluing, it is convenient to axiomatize the situation. Assume given a category I. Denote by \(I^<\) the category obtained by formally adding an initial object o to I, and denote by \(j:I \rightarrow I^<\) the natural embedding functor.
Definition 4.19
A 2limit diagram over I is a precofibration \(\mathcal {C}^< \rightarrow I^<\) such that for any precofibration \(\widetilde{\mathcal {C}} \rightarrow I^<\), any equivalence \(\rho :j^*\widetilde{\mathcal {C}} \rightarrow j^*\mathcal {C}^<\) over \(I \subset I^<\) extends to a functor \(\widetilde{\rho }:\widetilde{\mathcal {C}} \rightarrow \mathcal {C}^<\) over \(I^<\) cocartesian over maps \(o \rightarrow i\), \(i \in I\), and this extension is unique up to a unique isomorphism.
Lemma 4.20
For any precofibration \(\mathcal {C}\rightarrow I\) over a small category I, there exists a 2limit diagram \(\mathcal {C}^< \rightarrow I^<\) equipped with an equivalence \(j^*\mathcal {C}^< \cong \mathcal {C}\) over I. Moreover, any two such 2limit diagrams are equivalent over \(I^<\), and the equivalence is unique up to a unique isomorphism.
Proof
 (i)
an object \(\rho (i) \in \mathcal {C}_i\) for any object \(i \in I\), and
 (ii)
a morphism \(\rho (f):\rho (i') \rightarrow f_!\rho (i)\) for any morphism \(f:i \rightarrow i'\) in I,
Definition 4.21
For any precofibration \(\pi :\mathcal {C}\rightarrow I\) over a small category I with the 2limit diagram \(\mathcal {C}^< \rightarrow I^<\) provided by Lemma 4.20, the fiber \(\mathcal {C}^<_o\) over the initial object \(o \in I^<\) is denoted \({\text {Rec}}(I,\mathcal {C})\), where \({\text {Rec}}\) stands for “recollement”, and its objects are called cosections of the precofibration \(\pi \).
We note that this explicit can be easily modified to apply to an arbitrary covariant lax functor, nor only to the normalized ones. The resulting category of cosections would not satisfy any obvious universal property, but it will be welldefined. We will not do it in the general case. Instead, let us consider another very wellknown special case—namely, the case when I is the point category \(\mathsf{pt}\). Covariant lax functors from \(\mathsf{pt}\) to \({\text {Cat}}\) are known as comonads.
Recall that explicitly, a comonad on a category \(\mathcal {C}\) is a functor \(\Phi :\mathcal {C}\rightarrow \mathcal {C}\) equipped with two maps \(\varepsilon :\Phi \rightarrow \mathsf{Id}\), \(\mu :\Phi \rightarrow \Phi \circ \Phi \) such that \((\varepsilon \circ \mathsf{id}) \circ \mu = (\mathsf{id}\circ \varepsilon ) \circ \mu = \mathsf{id}\) and \((\mu \circ \mathsf{id}) \circ \mu = (\mathsf{id}\circ \mu ) \circ \mu \) (if \(\mathcal {C}\) is small, one can equivalently say that \(\Phi \) is a coalgebra in the monoidal category of endofunctors of \(\mathcal {C}\)). A standard source of comonads is adjunction: a functor \(\rho :\mathcal {C}\rightarrow \mathcal {E}\) to some category \(\mathcal {C}\) that admits a leftadjoint \(\lambda :\mathcal {E}\rightarrow \mathcal {C}\) induces a comonad structure on \(\Phi = \lambda \circ \rho \). Dually, \(\Phi _\dagger = \rho \circ \lambda \) is a monad on \(\mathcal {E}\) (that is, a comonad on the opposite category \(\mathcal {E}^o\)).
Example 4.22
Let \(\Delta \) be the category of finite ordinals, as in Example 4.7 and Remark 4.8, and let \(\Delta _+ \subset \Delta \) be the subcategory with the same objects and maps between them that send the initial element to the initial element. Then the embedding functor \(\rho :\Delta _+ \rightarrow \Delta \) admits a leftadjoint \(\lambda :\Delta \rightarrow \Delta _+\) that adds a new initial element to an ordinal. We thus obtain a comonad \(\varphi = \rho \circ \lambda \) on \(\Delta _+\). In this case, we have \({\text {coalg}}(\Delta _+,\varphi ) \cong \Delta \), and \({\text {Coalg}}(\Delta _+,\varphi ) \supset {\text {coalg}}(\Delta _+,\varphi ) \cong \Delta \) is the category \(\Delta ^<\) obtained by adding the new initial element o corresponding to the empty ordinal \(\emptyset \).
Example 4.23
Lemma 4.24
Proof
5 Simplicial replacements I
The categorical gluing constructions of Sect. 4.5 do not work too well in the homotopical setting: the associativity constraints involved in the definitions tend to produce higher order operations, and the whole assembly becomes very hard to control. Therefore we need another description of the categories \({\text {Rec}}(I,\mathcal {C})\) and \({\text {Coalg}}(\mathcal {C},\Phi )\). It turns out that this is possible if one uses simplicial methods, in the spirit of the treatment of monads and comonads in [37, VII.6]. In this section, we prepare the ground by introducing and studying a special class of prefibrations over the category \(\Delta \) of Example 4.22.
5.1 Special prefibrations
More generally, assume given a simplicial set \(X \in \Delta ^o{\text {Sets}}\), and denote by \(\Delta X\) its category of simplices—its objects are pairs \(\langle [n],x \rangle \), \([n] \in \Delta \), \(x \in X([n])\), and morphisms from \(\langle [n],x \rangle \) to \(\langle [n'],x' \rangle \) are morphisms \(f:[n] \rightarrow [n']\) such that \(f(x')=x\). The forgetful functor \(\pi :\Delta X \rightarrow \Delta \), \(\langle [n],x \rangle \mapsto [n]\) is the discrete fibration that corresponds to X by the Grothendieck construction, and every discrete fibration over \(\Delta \) with small fibers arises in this way.
Since \(\pi :\Delta X \rightarrow \Delta \) is a discrete fibration, lifting a morphism \(f:[n] \rightarrow [n']\) in \(\Delta \) to a morphism in \(\Delta X\) amounts to choosing a simplex \(x' \in X([n'])\), and lifting a commutative square (5.1) to a commutative square in \(\Delta X\) amounts to choosing \(x_{12} \in X([n_{12}])\). Say that such a lifted morphism f is special resp. cospecial if it is so as a morphism in \(\Delta \), and say such a lifted square is standard if it is standard in \(\Delta \).
Definition 5.1
 (i)
for any special map f in \(\Delta \), \(f^*\) is an equivalence of categories,
 (ii)
the natural map (4.2) is an isomorphism as soon as both \(f_1\) and \(f_2\) are either special or cospecial, and
 (iii)
for any standard square (5.1) with \(f=i_1 \circ p = i \circ p_1\), the maps (4.2) provide isomorphisms \(p^* \circ i_1^* \cong f^* \cong i^* \circ p_1^*\).
 (iv)
the natural map (4.2) is an isomorphism if \(\pi (f_1)\) is surjective.
Remark 5.2
Note that since \([0] \in \Delta _+ \subset \Delta \) is the initial object of the category \(\Delta _+\), for any simplicial set X and object \(c \in \Delta X\), there exists a unique special map \(s:c' \rightarrow c\) such that \(\pi (c') = [0] \in \Delta \). We will call such maps base special maps. Then Definition 5.1 (i),(ii) immediately show that it suffices to check Definition 5.1 (iii) for squares (5.1) such that p is a base special map.
Definition 5.3
 (i)
for any base special map \(f:c' \rightarrow c\) in \(\Delta X\), the corresponding transition functor \(f^*:\mathcal {C}'_c \rightarrow \mathcal {C}'_{c'}\) admits a fully faithful leftadjoint \(f_!:\mathcal {C}'_{c'} \rightarrow \mathcal {C}'_c\) with essential image \(\mathcal {C}_c \subset \mathcal {C}'_c\), and
 (ii)
for any cospecial map \(f:c \rightarrow c'\) in \(\Delta X\), the corresponding transition functor \(f^*:\mathcal {C}'_{c'} \rightarrow \mathcal {C}'_c\) sends \(\mathcal {C}_{c'} \subset \mathcal {C}'_{c'}\) into \(\mathcal {C}_c \subset \mathcal {C}'_c\).
Lemma 5.4
Assume given a simplicial set X and a generating fibration \(\mathcal {C}' \rightarrow \Delta X\), and let \(\mathcal {C}\subset \mathcal {C}'\) be the full subcategory spanned by the subcategories \(\mathcal {C}_c \subset \mathcal {C}'_c \subset \mathcal {C}'\), \(c \in \Delta X\). Then the projection \(\mathcal {C}\rightarrow \Delta \) is a special prefibration in the sense of Definition 5.1, and the embedding \(\mathcal {C}\rightarrow \mathcal {C}'\) is cartesian along cospecial and base special maps
Proof
The fact that \(\mathcal {C}\rightarrow \Delta \) is a prefibration and \(\mathcal {C}\rightarrow \mathcal {C}'\) is cartesian along base special maps immediately follows from Definition 5.3 (i), as in Example 4.9. Moreover, since \(f_!\) is fully faithful, the transition functor \(f^*:\mathcal {C}_c \rightarrow \mathcal {C}_{c'}\) is an equivalence for any base special map \(f:c' \rightarrow c\), and for any special map \(g:c \rightarrow c''\), we have \(g^* \circ f_! \cong (g \circ f)_!\). Thus \((g \circ f)^* \cong f^* \circ g^*\), so that \(f^*\) is an equivalence, and we have \((f_1 \circ f_2)^* \cong f_2^* \circ f_1^*\) for any composable pair of special maps \(f_1\), \(f_2\). This proves Definition 5.1 (i) and the special case of (ii). Then by Definition 5.3 (ii), for any cospecial map \(f:c \rightarrow c'\), the transition functor \(f^*:\mathcal {C}'_{c'} \rightarrow \mathcal {C}'_c\) of the fibration \(\mathcal {C}'\) restricts to the transition functor of the prefibration \(\mathcal {C}\). This means that \(\mathcal {C}\rightarrow \mathcal {C}'\) is cartesian along f. Moreover, since \(\mathcal {C}'\) is a fibration, this implies the remaining cospecial case of Definition 5.1 (ii), and Definition 5.1 (iii) for a square (5.1) with base special map p. By Remark 5.2, this is enough. \(\square \)
5.2 Canonical embedding
It turns out that Lemma 5.4 admits a converse: every special prefibration \(\mathcal {C}\rightarrow \Delta X\) is canonically generated by a generating fibration \(\mathcal {C}' \rightarrow \Delta X\). To prove this, we need one useful generalization of the simplicial replacement construction.
Say that a class of morphisms v in a category \(\mathcal {C}\) is multiplicative if it is closed under compositions and contains all identity maps, and assume given a category \(\mathcal {C}\) and two classes of morphisms v, c in \(\mathcal {C}\) such that v is multiplicative.
Definition 5.5

objects of \(\Delta ^c_v \mathcal {C}\) are pairs \(\langle [n],\kappa \rangle \) of an ordinal \([n] \in \Delta \) and a functor \(\kappa :[n] \rightarrow \mathcal {C}\) such that for any \(l \in [n]\), \(l \ge 1\), the map \(\kappa (l1) \rightarrow \kappa (l)\) is in the class c, and

morphisms from \(\langle [n],\kappa \rangle \) to \(\langle [n'],\kappa ' \rangle \) are pairs \(\langle f,b \rangle \) of a map \(f:[n] \rightarrow [n']\) in \(\Delta \) and a map \(b:\kappa \rightarrow \kappa ' \circ f\) such that for any \(i \in [n]\), the map \(b(i):\kappa (i) \rightarrow \kappa '(f(i))\) is in the class v.
Now assume given a simplicial set X and a special prefibration \(\pi :\mathcal {C}\rightarrow \Delta X\). Say that a map f in \(\mathcal {C}\) is special if \(\pi (f)\) is special, and say that f is cospecial if \(\pi (f)\) is cospecial and f is cartesian over \(\Delta \). Denote the classes of special resp. cospecial maps by s resp. c. Note that the class s is multiplicative, and by Definition 5.1 (ii), the same is true for the class c. Denote by \(\mathcal {C}_+,\mathcal {C}_ \subset \mathcal {C}\) the subcategories spanned by special resp. cospecial maps.
Lemma 5.6
Proof
Lemma 5.7
The fibration \(B(\mathcal {C}) \rightarrow \Delta X\) of (5.10) is generating in the sense of Definition 5.3, and the corresponding special prefibration over \(\Delta X\) is naturally equivalent to \(\mathcal {C}\).
Proof
5.3 Reflections
If a precofibration \(\mathcal {C}\rightarrow I\) over a small category I is in fact a cofibration, then the pullback \(\xi ^*\mathcal {C}^\perp \) of the transpose fibration \(\mathcal {C}^\perp \) to the simplicial replacement \(\Delta I\) is special, and it is not difficult to check that the pullback functor (5.5) identifies the categories \({\text {Rec}}(I,\mathcal {C}) \cong {\text {Sec}}(I,\mathcal {C}^\perp )\) and \({\text {Sec}}_+(\Delta I,\xi ^*\mathcal {C}^\perp )\). However, note that we also have the functor \(\beta :\Delta I \rightarrow I\), and we can consider the fibration \(\beta ^{o*}\mathcal {C}^\perp \) on the category \((\Delta I)^o\). This gives the same result: cosections of \(\mathcal {C}\) correspond to sections of \(\beta ^{o*}\mathcal {C}^\perp \) of a certain special type.
It turns out that for any simplicial set X, there exists a correspondence between prefibrations over \(\Delta X\) and \((\Delta X)^o\) and their sections, and prefibrations over \((\Delta X)^o\) are in fact more convenient for homotopical applications. Let us present this correspondence.
Consider again the category \(\Delta _\flat \) of Example 4.7, with functors \(\nu \), \(\sigma \), \(\sigma _\dagger \), and the corresponding category \(\Delta _\flat X = \Delta _\flat \times _\Delta \Delta X\), with the functors \(\nu \), \(\sigma \) and \(\sigma _\dagger \) of (5.9), and the discrete fibration \(\pi :\Delta _\flat X \rightarrow \Delta _\flat \).
Definition 5.8
 (i)
the transition functor \(f^*\) is an equivalence for any special f, and
 (ii)
the natural map (4.2) is an isomorphism as soon as \(f_2\) is special, or \(f_1\) is cospecial, or \(f_1\) is special, \(f_2\) is cospecial, and \(f_1 \circ f_2\) is perfect.
Lemma 5.9
Proof
The functor \(\sigma \) sends special maps to special maps, so that for any special \(\mathcal {C}\rightarrow \Delta _\flat X\), \(\sigma ^*\mathcal {C}\) satisfies Definition 5.1 (i) and (ii) if \(f_1\) and \(f_2\) are special. For the rest, note that we still have the unique decomposition (5.2) with special i and cospecial p for any map f in both \(\Delta X\) and \(\Delta _\flat X\). Then for any cospecial map \(f_1\) in \(\Delta X\) with such decomposition \(\sigma (f) = p \circ i\), we have \(f^* \cong i^* \circ p^*\), and for two composable maps \(f_1\), \(f_2\) with \(\sigma (f_1)=p_1 \circ i_1\), \(\sigma (f_2) = p_2 \circ i_2\), we have \(f_2^* \circ f_1^* \cong i_2^* \circ p_2^* \circ i_1^* \circ p_1^*\). But \(i_1 \circ p_2\) is perfect, so this is isomorphic to \(\sigma (f_1 \circ f_2)^*\). Thus \(\sigma ^*\mathcal {C}\) satisfies Definition 5.1 (ii), and to check (iii), it remains to observe that for any standard square (5.1) in \(\Delta X\), \(\sigma (p_2 \circ i)\) is a perfect map.
Analogously, \(\sigma _\dagger \) sends special maps to special maps, so that for any special \(\mathcal {C}\rightarrow \Delta X\), \(\sigma _\dagger ^\mathcal {C}\) satisfies Definition 5.8 (ii). Moreover, \(\sigma _\dagger \) also sends cospecial maps to cospecial maps, so to prove that \(\sigma _\dagger ^*\mathcal {C}\) is special, it suffices to check that (4.2) is an isomorphism when either \(f_1\) is cospecial, \(f_2\) special, or \(f_1\) is special, \(f_2\) is cospecial, and the composition \(f_1 \circ f_2\) is perfect. In both cases, \(\sigma _\dagger (f_1)\) and \(\sigma _\dagger (f_2)\) fit into a standard square, so the claim follows from Definition 5.1 (iii).
For the second claim, note that for any \(x \in \Delta _\flat X\), the adjunction map \(a_x:\sigma _\dagger (\sigma _(x)) \rightarrow x\) is special. Moreover, for any map \(f:x\rightarrow x'\) with decomposition \(\sigma _\dagger (\sigma (f)) = p \circ i\) of (5.2), the composition map \(a_{x'} \circ p\) is perfect. Therefore the adjunction map \(a:\sigma _\dagger \circ \sigma \rightarrow \mathsf{id}\) is compatible with \(\mathcal {C}\) in the sense of Definition 4.17, and it becomes strictly compatible when we restrict to the subcategory spanned by special maps. Thus we have a functor \(a^*:\mathcal {C}\rightarrow \sigma _\dagger ^*\sigma ^*\mathcal {C}\), this functor is an equivalence by Definition 5.8 (i), and this equivalence induces an equivalence between the categories of special sections. \(\square \)
Definition 5.10
A map f in \((\Delta X)^o\) is special resp. cospecial if \(\iota ^o(f^o)\) is special resp. cospecial in \(\Delta X\). A map f in \((\Delta _\flat X)^\perp \) is special resp. cospecial resp. perfect if \(\iota _\flat ^{1}(f^o)\) is special resp. cospecial resp. perfect in \(\Delta _\flat X\). A prefibration \(\mathcal {C}\) over \((\Delta x)^o\) is special if \(f^*\) is an equivalence for any special map f, (4.2) is an isomorphism if both \(f_1\) and \(f_2\) are special or cospecial, and for any standard square in \(\Delta X\) with \(f=i_1 \circ p = i \circ p_1\), the maps (4.2) provide isomorphisms \(\iota (p_1^o)^* \circ \iota (i^o)^* \cong \iota (f^o)^* \cong \iota (i^o_1)^* \circ \iota (p^o)^*\). It is normalized if in addition, (4.2) is an isomorphism as soon as \(\pi (f_2)\) is injective in \(\Delta ^o\). A prefibration \(\mathcal {C}\) over \((\Delta _\flat X)^\perp \) is special if \(f^*\) is an equivalence for any special f, and (4.2) is an isomorphism as soon as \(f_1\) is special, or \(f_2\) is cospecial, or \(f_2\) is special, \(f_1\) is cospecial, and \(f_1 \circ f_2\) is perfect. As in Definition 5.1, a section of a special prefibration is special if it is cartesian along all special maps, and \({\text {Sec}}_+ \subset {\text {Sec}}\) denotes the full subcategory spanned by special sections.
Then by the same argument as in Lemma 5.9, the pullback operations \(\tau ^*\), \(\tau _\dagger ^*\) send special prefibrations to special prefibrations. Moreover, the map \(a_\dagger :\mathsf{Id}\rightarrow \tau \circ \tau _\dagger \) is pointwise special, thus compatible with any special prefibration \(\mathcal {C}\) over \((\Delta _\flat X)^\perp \), and strictly compatible if we restrict to the subcategory spanned by special maps. Therefore as in Lemma 5.9, \(a_\dagger ^*:\tau ^*\tau _\dagger ^*\mathcal {C}\rightarrow \mathcal {C}\) is an equivalence for any special \(\mathcal {C}\), and this equivalence identifies the categories of special sections.
Proposition 5.11
Proof
For the first claim, note that since \(\mathcal {C}\) is special, \(\sigma _\dagger ^*\mathcal {C}\) is special by Lemma 5.9. Then \(\sigma _\dagger ^*\mathcal {C}\rightarrow \Delta X\) is a precofibration by Definition 5.8 and Lemma 4.12, and by Definition 5.1 (ii), it is actually a cofibration. Thus the transpose functor \((\sigma ^*_\dagger \mathcal {C})^\perp \rightarrow (\Delta _\flat X)^\perp \) is welldefined. It is obviously a prefibration, with the transition functor corresponding to a map represented by a diagram (4.20) given by \(v^* \circ c_!\), and the conditions of Definition 5.10 reduce to the corresponding conditions for \(\sigma _\dagger ^*\mathcal {C}\). Thus \((\sigma _\dagger ^*\mathcal {C})^\perp \) is special, and then so is its restriction \(\mathcal {C}^\flat \).
Dually, \(\tau _\dagger ^*\mathcal {C}' \rightarrow (\Delta X)^o\) is a prefibration by virtue of Definition 5.1 (ii) and Lemma 4.11, and Definition 5.1 (ii) then shows that it is actually a fibration, and \(\tau _\dagger ^*\mathcal {C}' \rightarrow (\Delta _\flat X)^\perp \) is cartesian. Then \((\tau _\dagger ^*\mathcal {C}')_\perp \) is welldefined, and again, it is obviously a prefibration with transition functors \(c_! \circ v^*\). The conditions of Definition 5.8 reduce to those of Definition 5.10, so that the prefibration is special, and so is its restriction \(\mathcal {C}_\flat \).
Finally, the equivalences \((\mathcal {C}^\flat )_\flat \cong \mathcal {C}\), \((\mathcal {C}'_\flat )^\flat \cong \mathcal {C}\) immediately follow from (4.21), and the equivalences (5.15) then follows from Lemma 5.9 and its counterpart for \((\Delta X)^o\). \(\square \)
Definition 5.12
For any special prefibration \(\mathcal {C}\) over \(\Delta X\) resp. \((\Delta X)^o\), the special prefibration \(\mathcal {C}^\flat \) resp. \(\mathcal {C}_\flat \) of (5.13) resp. (5.14) is called the reflection of the prefibration \(\mathcal {C}\). For any special section \(E \in {\text {Sec}}_+(\Delta X,\mathcal {C})\) resp. \(E \in {\text {Sec}}_+((\Delta X)^o,\mathcal {C})\), its reflection \(E^\flat \) resp. \(E_\flat \) is the corresponding special section of \(\mathcal {C}^\flat \) resp. \(\mathcal {C}_\flat \).
One immediate application of reflections is a version of the identification (4.11) relative over \(\Delta X\). Namely, let \(\Delta ^\flat _\flat \subset \Delta ^\flat \times _\Delta \Delta _\flat \) be the category of triples \(\langle [n],p,q \rangle \), \([n] \in \Delta \), \(p,q \in [n]\), \(p \le q\), with maps from \(\langle [n],p,q \rangle \) to \(\langle [n'],p',q' \rangle \) given by maps \(f:[n] \rightarrow [n']\) such that \(f(p) \ge p'\) and \(f(q) \le q'\). The forgetful functor \(\nu :\Delta ^\flat _\flat \rightarrow \Delta \) is a cofibration, with the fully faithful leftadjoint functor \(\varepsilon :\Delta \rightarrow \Delta ^\flat _\flat \) sending [n] to \(\langle [n],0,n \rangle \). Moreover, sending \(\langle [n],p,q \rangle \) to \(\langle [n],p \rangle \) resp. \(\langle [n],q \rangle \) gives a projection from \(\Delta ^\flat _\flat \) to the category \(\Delta _\flat \) resp. \(\Delta ^\flat \cong (\Delta ^\perp _\flat )^o\), and composing these projections with \(\sigma _\dagger \) resp. \(\tau _\dagger ^o\), we obtain two projections \(\widetilde{\sigma },\widetilde{\tau }:\Delta ^\flat _\flat \rightarrow \Delta \).
Lemma 5.13
Proof
Since \(\varepsilon :\Delta X \rightarrow \Delta ^\flat _\flat X\) admits a rightadjoint functor, \(\mathsf{lim}_{\Delta X}\varepsilon ^*F \cong \mathsf{lim}_{\Delta ^\flat _\flat X}F\) for any \(F:\Delta ^\flat _\flat X \rightarrow {\text {Sets}}\), so we have the second identification in (5.18). For the first one, note that by (4.11), an element in the limit defines a map \(\tau _\dagger ^*A^\flat _{\langle [n],x \rangle } = \sigma _\dagger ^*A_{\langle [n],x \rangle } \rightarrow \sigma _\dagger ^*B_{\langle [n],x \rangle }\), and these maps patch together into a single map \(\sigma _\dagger ^*A \rightarrow \sigma _\dagger ^*B\) if and only if they are compatible with the maps (5.17). \(\square \)
6 Simplicial replacements II
We can now use the technology of special prefibrations to describe the gluing constructions of Sect. 4.5 purely in terms of sections. First, we do it in the case of the trivial base category \(I=\mathsf{pt}\) and nonnormalized lax functors. In Sect. 6.1, we associate a special prefibration over \(\Delta \) to a comonad and construct a functor from the category of coalgebras over a comonad to the category of sections of the prefibration (this is Lemma 6.4). Then in Sect. 6.2, we show that this is in fact a onetoone correspondence between comonads and special prefibrations over \(\Delta \) (Proposition 6.7) that identifies special sections and coalgebra over the comonad (Lemma 6.6). In Sect. 6.3, we turn to normalized lax functors—that is, precofibrations—over an arbitrary I. The main result is Proposition 6.11 that establishes a onetoone correspondence between precofibrations over I and normalized special fibrations over its simplicial replacement \(\Delta I\), and shows that this correspondence also identifies cosections of the precofibration and special sections of the special prefibration. Finally, in Section 6.6, we illustrate our abstract canonical construction by explicit expressions in terms of the lax functors.
6.1 Barycentric expansions
We start with comonads. We will show that coalgebras over a comonad naturally appear as sections of a special prefibration over the category \(\Delta \).
For any map \(f:[n_1] \rightarrow [n_2]\) in \(\Delta \), we have the obvious pullback functor \(f^*:F^{[n_2]} \rightarrow F^{[n_1]}\), and these are compatible with compositions, so that the categories \(F^{[n]}\) form a fibration \(S(F) \rightarrow \Delta \) with fibers \(S(F)_{[n]} = F^{[n]}\) and transition functors \(f^*\).
Now recall that we have the subcategory \(\Delta _+ \subset \Delta \), and by definition, \([0] \in \Delta _+\) is the initial object, so that for any \([n] \in \Delta _+ \subset \Delta \) we have a unique map \(s:[0] \rightarrow [n]\).
Lemma 6.1
Assume that the functor F has a leftadjoint \(F_\dagger :\mathcal {C}_1 \rightarrow \mathcal {C}_0\). Then for any \([n] \in \Delta _+ \subset \Delta \), the pullback functor \(s^*:F^{[n]} \rightarrow F^{[0]}\) admits a fully faithful leftadjoint \(s_!\). Moreover, an object \(\langle \kappa ,\{c_\cdot \}\rangle \in F^{[n]}\) lies in the essential image of the functor \(s_!\) iff for any i, \(0 \le i < n\), the map \(F_\dagger (\kappa (i)) \rightarrow c_{i+1}\) adjoint to the map \(\kappa (i) \rightarrow \kappa (i+1) \cong F(c_{i+1})\) is an isomorphism.
Proof
For any \([n] \in \Delta \), denote by \(s_n:[n] \rightarrow [n+1]\) the embedding onto the initial segment of the ordinal \([n+1]\) (that is, \(s_n(l) = l\), \(0 \le l \le n\)). Then it is immediately obvious that \(s_n^*\) does have a leftadjoint \(s_{n!}\): it extends a pair \(\langle \kappa ,\{c_\cdot \}\rangle \) by setting \(c_{n+1} = F_\dagger (\kappa (n))\), \(\kappa (n+1)=F(c_{n+1})\), and letting the map \(\kappa (n) \rightarrow \kappa (n+1)=F(c_{n+1})\) be adjoint to the isomorphism \(F_\dagger (\kappa (n)) \cong c_{n+1}\). Since \(s_n^* \circ s_{n!} \cong \mathsf{Id}\) by construction, \(s_{n!}\) is fully faithful, and since the map \(s:[0] \rightarrow [n]\) is the composition of embeddings \(s_l\), \(0 \le l < n\), it has a full faithful leftadjoint by induction, and the same induction describes its essential image. \(\square \)
Now assume given a comonad \(\Phi \) on a category \(\mathcal {C}\), and let \(F:\mathcal {C}\rightarrow {\text {Coalg}}(\Phi ,\mathcal {C})\) be the free coalgebra functor. By definition, it does have a leftadjoint given by the forgetful functor, so we are within the framework of Lemma 5.4.
Definition 6.2
The barycentric expansion \(S(\mathcal {C},\Phi )\) of the comonad \(\Phi \) is the full subcategory \(S(\mathcal {C},\Phi ) \subset S(F)\) spanned by the essential images of the fully faithful embeddings \(s_!:\mathcal {C}\cong F^{[0]} \rightarrow F^{[n]}\) of Lemma 6.1.
Remark 6.3
Note that for any functor \(\widetilde{F}:\mathcal {C}\rightarrow \widetilde{\mathcal {C}}\) that has a leftadjoint \(\widetilde{F}_\dagger \) such that \(\Phi \cong \widetilde{F}_\dagger \circ \widetilde{F}\) as a comonad, and for any \([n] \in \Delta \), the comparison functor (4.33) induces a functor \(\widetilde{F}^{[n]} \rightarrow F^{[n]}\). Moreover, it is immediately obvious from the definitions that this functor is an equivalence that identifies the essential images of the fully faithful embeddings \(s_!\) of Lemma 5.4. Therefore to define the barycentric subdivision, we can replace \({\text {Coalg}}(\Phi ,\mathcal {C})\) with \(\widetilde{\mathcal {C}}\) and F with \(\widetilde{F}\) without changing the result.
By Lemma 6.1, the subcategory \(S(\mathcal {C},\Phi ) \subset S(F)\) is as in Example 4.9, so that the projection \(S(\mathcal {C},\Phi ) \rightarrow \Delta \) is a prefibration. If \(\Phi = \mathsf{Id}\) is the trivial comonad, then \(S(\mathcal {C},\Phi ) \cong \mathcal {C}\times \Delta \) is the trivial prefibration with fibers \(\mathcal {C}= S(\mathcal {C},\Phi )_{[0]}\). In the general case, the fibers are the same, but the transition functors are nontrivial and depend on \(\Phi \).
Lemma 6.4
Proof
For the first claim, it suffices to check that \(S(\mathcal {C},\Phi )\) satisfies the conditions of Lemma 5.4, and this immediately follows from Lemma 6.1. For the second claim, it suffices to check that the functor (6.3) is cartesian along special maps, and since \(S(\tau )\) is cartesian, this amounts to checking that \(S(\sigma )_\dagger \) is cartesian along special maps. Indeed, by definition the maps (4.5) for this functor are adjoint to the corresponding maps for the cartesian functor \(S(\sigma )\), and since \(f^*\) is an equivalence for a special map f, these maps are isomorphisms. \(\square \)
6.2 Comonads and coalgebras
Lemma 6.5
Assume given a special prefibration \(\mathcal {C}\) over \(\Delta X\) in the sense of Definition 5.1, and let \(\mathcal {C}' = a_X^*\mathcal {C}\) be its preimage under the functor (6.6). Then the functor \(\rho _X\) of (6.7) and the prefibration \(\mathcal {C}'\) satisfy the assumptions of Lemma 4.18, so that the pullback functor \(\rho _X^*\) admits a leftadjoint functor \(\lambda ^{\mathcal {C}'}:{\text {Sec}}(\Delta _+X,\rho _X^*\mathcal {C}') \rightarrow {\text {Sec}}(\Delta _<X,\mathcal {C}')\). Moreover, \(\lambda ^{\mathcal {C}'}\) sends cartesian sections to special sections, and if \(\mathcal {C}\) is normalized, then \(\lambda ^{\mathcal {C}'}\) is fully faithful.
Proof
The leftadjoint to \(\rho _X\) is the tautological embedding \(\lambda \), and the fact that the adjunction map \(a:\mathsf{id}\rightarrow \rho _X \circ \lambda \) is compatible with \(\mathcal {C}'\) immediately follows from Definition 5.1 (iii). It also follows from Definition 5.1 that a becomes strictly compatible with \(\mathcal {C}'\) after restricting to \(\Delta _+X\), so that \(\lambda ^{\mathcal {C}'}\) sends cartesian sections to special sections. Moreover, the adjunction map \(\mathsf{Id}\rightarrow \rho _X^* \circ \lambda ^{\mathcal {C}'}\) is induced by the map \(\alpha (a_\dagger ,a)\) of (4.2) for \(\mathsf{Id}= a_\dagger \circ a\), and since \(a_\dagger \) is surjective, this is an isomorphism if \(\mathcal {C}\) is normalized. \(\square \)
Lemma 6.6
For any comonad \(\Phi '\) on a category \(\mathcal {C}\), with barycentric expansion \(S(\mathcal {C},\Phi ')\), the comonad \(\Phi \) of (6.12) on \(\mathcal {C}= S(\mathcal {C},\Phi ')_{[0]}\) is canonically isomorphic to \(\Phi '\), and the functor (6.5) is an equivalence of categories inverse to (6.13).
Proof
Proposition 6.7
Proof
6.3 Cosections
Now assume given a small category I, with its simplicial replacement \(\Delta I = \Delta NI\), and a precofibration \(\mathcal {C}\rightarrow I\). Let \(\widetilde{S}(\mathcal {C}) = \Delta _v\mathcal {C}\), and let \(S(\mathcal {C}) = \Delta ^c_v \mathcal {C}\subset \widetilde{S}(\mathcal {C})\), where c resp. v are the classes of cocartesian resp. vertical maps in \(\mathcal {C}\). The precofibration \(\mathcal {C}\rightarrow I\) then induces a functor \(\widetilde{S}(\mathcal {C}) \rightarrow \Delta I\) that restricts to a functor \(S(\mathcal {C}) \rightarrow \Delta I\).
Lemma 6.8
The functor \(S(\mathcal {C}) \rightarrow \Delta I\) is a normalized special prefibration in the sense of Definition 5.1.
Proof
Since the class of all maps is closed under compositions, the projection \(\widetilde{\mathcal {C}} \rightarrow \Delta \) is a fibration, and since \(\Delta I \rightarrow I\) is a discrete fibration, \(\widetilde{S}(\mathcal {C}) \rightarrow \Delta I\) is a fibration as well (a cartesian lifting of a map is given by the cartesian lifting of its image in \(\Delta \)). By definition, we have a full embedding \(S(\mathcal {C}) \subset \widetilde{S}(\mathcal {C})\), and it immediately follows from the definition of a cocartesian map that the fibration \(\widetilde{S}(\mathcal {C}) \rightarrow \Delta I\) generating of Definition 5.3, and \(S(\mathcal {C}) \subset \widetilde{S}(\mathcal {C})\) is the associated special prefibration. Finally, since identity maps are always cocartesian, for any surjective map \(f:[n'] \rightarrow [n]\) and \(\kappa :[n] \rightarrow I\), the corresponding functor \(f^*\) of the fibration \(\widetilde{S}(\mathcal {C})\) sends \(S(\mathcal {C})_{[n,\kappa ]} \subset \widetilde{S}(\mathcal {C})_{[n,\kappa ]}\) into \(S(\mathcal {C})_{[n',f^*\kappa ]} \subset \widetilde{S}(\mathcal {C})_{[n',f^*\kappa ]}\). Therefore the special prefibration \(S(\mathcal {C}) \rightarrow \Delta I\) is normalized. \(\square \)
Definition 6.9
The special prefibration \(S(\mathcal {C}) \rightarrow \Delta I\) is called the barycentric expansion of the precofibration \(\mathcal {C}\rightarrow I\).
 (i)
objects are pairs \(\langle i,s \rangle \) of an object \(i \in I\) and a special spection \(s \in {\text {Sec}}_+(\mathsf{L}(\xi )_i,\sigma _i^*\mathcal {C})\),
 (ii)
morphisms from \(\langle i,s \rangle \) to \(\langle i',s' \rangle \) are pairs \(\langle f,\alpha \rangle \) of a morphism \(f:i \rightarrow i'\) and a morphism \((f^o_!)^*s \rightarrow s'\).
Definition 6.10
The reduction \(R(\mathcal {C})\) of the normalized special prefibration \(\mathcal {C}\) is the full subcategory \(R(\mathcal {C}) \subset \widetilde{R}(\mathcal {C})\) spanned by the essential images of the fully faithful functors (6.20).
Note that by Lemma 6.5, the functors \(\lambda ^\mathcal {C}_i\) of (6.20) are leftadjoint to the pullback functors \(\rho _i^*\), so that as in Example 4.9, the projection \(R(\mathcal {C}) \rightarrow I\) induced by the cofibration \(\widetilde{R}(\mathcal {C}) \rightarrow I\) is a precofibration.
Proposition 6.11
Proof
6.4 Explicit formulas
In principle, one can combine Propositions 6.11 and 6.7 to obtain a bijective correspondence between special prefibrations over the simplicial replacement \(\Delta I\) of a small category I and covariant lax functors F from I to \({\text {Cat}}\) in the sense of Definition 4.5. Proposition 6.11 resp. Proposition 6.7 would then correspond to the special cases when F is normalized resp. \(I=\mathsf{pt}\) is a point. Unfortunately, in the general case, there is no easy way to characterize lax functors invariantly, so that the correspondence has to be constructed and proved by hand. We will not do it. However, it is perhaps useful to at least describe the correspondence (this would also work in the situation of Propositions 6.11 and 6.7, where we did give invariant proofs).
For the reflected prefibration \(\widetilde{F}^\flat \), the formulas are essentially the same, but we have to count from the other end of a diagram. The fiber \(\widetilde{F}^\flat _{\langle [n],i_\cdot \rangle }\) is now given by \(\widetilde{F}^\flat _{\langle [n],i_\cdot \rangle } = F(i_n)\), and the transition functor \(g^*\) is given by \(g^* = F(i^\flat (g)_\cdot )\), where \(i^\flat (g)_\cdot \) is the terminal segment \(i_{g(n')} \rightarrow \dots \rightarrow i_n\) of the diagram \(i_\cdot \), of length \(n  g(n')\). The map (4.2) is again given by \(\alpha (c^\flat (g_1,g_2))\), where \(c^\flat (g_1,g_2):i^\flat (g_2)_\cdot * i^\flat (g_1)_\cdot \rightarrow i^\flat (g_1 \circ g_2)\) is the natural map.
Another way to package the same data is by introducing the path 2category PI of the category I. Its objects are objects \(i \in I\), 1morphisms from i to \(i'\) are pairs \(\langle [n],i_\cdot \rangle \) with \(i_0=i\) and \(i'=i_n\) understood as paths from i to \(i'\), with composition given by concatenation, and 2morphisms are maps \(g:\langle [n'],i'_\cdot \rangle \rightarrow \langle [n],i_\cdot \rangle \) such that \(g(0)=0\) and \(g(n')=n\). Then a lax functor F defines an honest 2functor \(PI \rightarrow {\text {Cat}}\), \(i \mapsto F(i)\), \(i_\cdot \mapsto F(i_\cdot )\), \(g \mapsto \alpha (g)\), and sending \(\langle [n],i_\cdot \rangle \) to \(i_0\) resp. \(i_1\) and g to \(i(g)_\cdot \) resp. \(i^\flat (g)_\cdot \) gives colax functors from \(\Delta I\) resp. \((\Delta I)^o\) to PI.
7 Reedy categories I
7.1 Ordered categories
Producing model structures on categories of sections of Grothendieck prefibrations is usually a highly nontrivial exercise. This is true even for trivial prefibrations: if we are given a relative category \(\langle \mathcal {C},W \rangle \) and a small category I, then in general, there is no known construction that would produce a model structure on \(\langle \mathcal {C}^I,W^I \rangle \) starting from a model structure on \(\langle \mathcal {C},W \rangle \). However, there are constructions that work either for a restricted class of model structures, or for a restricted class of small categories. Let us recall the latter. We start with a simple special case.
Definition 7.1
 (i)
A good filtration on a small category I is a collection of full subcategories \(I_{\le n} \subset I\), one for any nonnegative integer n, such that \(I_{\le n} \subset I_{\le n+1}\) and \(I = \bigcup I_{\le n}\).
 (ii)A small category I with a good filtration is ordered if
 (a)
for any integer \(n \ge 0\), the full subcategory \(I_n \subset I\) spanned by objects in \(I_{\le n}\) but not in \(I_{\le n1}\) is discrete (that is, all maps are identity maps),
 (b)
for any morphism \(f:i \rightarrow i'\), \(i \in I_{\le n}\) implies \(i' \in I_{\le n}\), and
 (c)
for any object \(i \in I\), the category I(i) of objects \(i' \in I\) equipped with a map \(f:i \rightarrow i'\) is finite.
 (a)
An equivalent way of giving a good filtration is to give a “degree function” \(\deg \) that associates a nonnegative integer n to any object \(i \in I\), so that \(I_{\le n}\) is the full subcategory spanned by objects i of degree \(\deg (i) \le n\).
Example 7.2
Let V(X) be the partially ordered set of finite subsets in a set X, considered as a small category in the usual way. Then the opposite category \(V(X)^o\) is ordered, with the degree function given by cardinality.
Lemma 7.3
Assume given a finite ordered category I and a model category \(\mathcal {C}\). Then \(\mathcal {C}^I\) has an injective model structure in the sense of Definition 2.7.
Sketch of a proof
\(\square \)
Remark 7.4
Passing to the opposite categories, we also see that in the assumptions of Lemma 7.3, \(\mathcal {C}^{I^o}\) has a projective model structure in the sense of Definition 2.7.
We also record right away a couple of properties of injective model structures provided by Lemma 7.3.
Lemma 7.5
 (i)
For any model category \(\mathcal {C}\) and object \(i \in I\), the functor \(\mathcal {C}^I \rightarrow \mathcal {C}\) given by evaluation at i is rightderivable with respect to the injective model structure on \(\mathcal {C}^I\).
 (ii)
For any model categories \(\mathcal {C}_1\), \(\mathcal {C}_2\) and a rightderivable functor \(\Phi :\mathcal {C}_1 \rightarrow \mathcal {C}_2\) that preserves finite limits, the functor \(\mathcal {C}_1^I \rightarrow \mathcal {C}_2^I\) induced by \(\Phi \) is rightderivable,
Proof
Both claims immediately follow from the inductive description of the injective model structure on \(\mathcal {C}^I\) given in Lemma 7.3 and the corresponding properties of the model structure of Proposition 3.8 (i). \(\square \)
We note that in Lemma 7.5 (i), the evaluation functor is also trivially leftderivable, and that in Lemma 7.5 (ii), it is essential to require that \(\Phi \) preserves finite limits.
7.2 Reedy categories
Definition 7.1 and Lemma 7.3 have a remarkable generalization. To state it, recall that a factorization system on a category \(\mathcal {C}\) consists of two classes of maps \(\langle L,M \rangle \) in \(\mathcal {C}\) that are closed under compositions and contain all identity maps, such that any morphism f in \(\mathcal {C}\) factors as \(f = m \circ l\), \(l \in L\), \(m \in M\), and such a factorization is unique up to a unique isomorphism. For example, for any simplicial set X, special and cospecial maps in \(\Delta ^o X\) in the sense of Definition 5.1 form a factorization system; for more details on this notion, see [8]. For any factorization system \(\langle L,M \rangle \) on a category \(\mathcal {C}\), we will denote by \(\mathcal {C}_L\) resp. \(\mathcal {C}_M\) the categories with the same objects as \(\mathcal {C}\) and morphisms that are in L resp. M.
Definition 7.6
A Reedy category is a small category I equipped with a factorization system \(\langle L,M \rangle \) and a good filtration in the sense of Definition 7.1 (i) such that both \(I_M\) and \(I^o_L\) with the induced filtrations are ordered in the sense of Definition 7.1 (ii).
We note that the notion of a Reedy category is selfdual—for any Reedy category I, the opposite category \(I^o\) is also a Reedy category, with the same good filtration and \(M^o\) resp. \(L^o\) as L resp. M. We also note that for any category \(\mathcal {C}\), we have \(\left( \mathcal {C}^{I_L}\right) ^o \cong \mathcal {C}^{oI^o_L}\), so that if \(\mathcal {C}\) is a model category, then Lemma 7.3 provides natural model structures both on \(\mathcal {C}^{I_M}\) and on \(\mathcal {C}^{I_L}\).
Example 7.7
The category \(\Delta \) is a Reedy category, with the degree function given by \(\deg ([n])=n\), and L resp. M consisting of injective resp. surjective maps.
Example 7.8
More generally, consider the category of simplices \(\Delta X\) of a simplicial set X, with the degree function \(\deg (\langle [n],x \rangle ) = n\), and a map f in L resp. M if it is in L resp. M after projecting to \(\Delta \). Then \(\Delta X\) is a Reedy category. The only nontrivial thing to check is Definition 7.1 (ii)(c) for the category \((\Delta X)_M\); this immediately follows from the fact that for any object \(\langle [n],x \rangle \in \Delta \), a surjective map \(f:[n] \rightarrow [n']\) in \(\Delta \) admits at most one lifting to a map \(\langle [n],x \rangle \rightarrow \langle [n'],x' \rangle \) in \(\Delta X\).
Example 7.9
The category \(\Gamma \) of finite sets \(\{0,\dots ,n\}\) and all maps between them is not a Reedy category. Indeed, Definition 7.1 (ii) (a) together with the factorization property imply that \({{\text {Aut}}}(i) = \{\mathsf{id}\}\) for any object \(i \in I\) in a Reedy category I.
Theorem 7.10
Assume given a Reedy category I and a relative category \(\langle \mathcal {C},W \rangle \) equipped with a model structure \(\langle C,F \rangle \). Denote by \(\lambda :I_L \rightarrow I\), \(\mu :I_M \rightarrow I\) the embedding functors, let \(\widetilde{C}\) be class of maps f in \(\mathcal {C}^I\) such that \(\lambda ^*(f)\) is a cofibration with respect to the projective model structure of Definition 2.7, and let \(\widetilde{F}\) be the class of maps f such that \(\mu ^*(f)\) is a fibration with respect to the injective model structure of Definition 2.7. Then \(\langle \widetilde{C},\widetilde{F} \rangle \) is a model structure on \(\langle \mathcal {C}^I,W^I \rangle \). \(\square \)
In the literature, Theorem 7.10 is usually formulated as a theoremconstruction without first stating Lemma 7.3 (which then appears as a special case, since every ordered category is trivially a Reedy category). While the theorem is due to Reedy [40], Definition 7.6 is not—[40] only treats the category \(\Delta \) of Example 7.7 and its opposite \(\Delta ^o\). The observation that exactly the same proof works in a larger generality was made later, and it is hard to trace when; it definitely appears in [26], a great reference for the subject, with further references to a book in preparation by Dwyer, Hirshhorn and Kan that, sadly, never appeared (although [14] uses Reedy model structures in a very essential way).
Example 7.11
Here are two very basic examples of how one uses Theorem 7.10, both taken out of [26]. Firstly, \([1] = \{0,1\}\) is a Reedy category in two different ways: one can either set \(\deg (0)=0\), \(\deg (1)=1\), or the other way around, \(\deg (0)=1\) and \(\deg (1)=0\). The two model structures one obtains are the injective and the projective model structure of Definition 2.7. Secondly, for the category \(\mathsf{V}\) of Sect. 2.2, there are four options: \(\deg (0)=0\), \(\deg (1)=\deg (2)=1\), or \(\deg (0)=1\), \(\deg (1)=\deg (2)=0\), or \(\deg (1)=2\), \(\deg (0)=1\), \(\deg (2)=0\), or finally \(\deg (1)=0\), \(\deg (0)=1\), \(\deg (2)=2\). The first two options again give the injective and projective model structures, and these give rise to the description of homotopy cartesian and homotopy cocartesian squares in terms of fiber and cofiber squares of Definition 2.8. The other two model structures on \(\mathcal {C}^V\) are new. Using these structures in the Quillen Adjunction Theorem, one can prove, for example, that a cartesian square (2.4) with fibrant \(Y'\) is homotopy cartesian as soon as either \(f'\) or \(g'\) is a fibration, and dually for cocartesian squares.
With this fact, the proof of Theorem 7.10 proceeds by induction on the degree, as in Lemma 7.3. In our context, one can easily deduce from (7.2) that the latching functors \(L_n\) resp. \(M_n\) are left resp. rightderivable, and then it suffices to apply Proposition 3.12 (i). The usual proofs spell things out more explicitly (and in fact, our proof of Proposition 3.12 is extracted from the standard proofs of Theorem 7.10).
7.3 Balzin theorem
A very useful result discovered recently by Balzin [1] generalizes Theorem 7.10 to nontrivial families of model categories.
Definition 7.12
 (i)
for any morphism \(l:i' \rightarrow i\) in L, the transition functor \(l^*:\mathcal {C}_i \rightarrow \mathcal {C}_{i'}\) has a leftadjoint \(l_!:\mathcal {C}_{i'} \rightarrow \mathcal {C}_i\), and
 (ii)
if we are given another morphism \(f:i \rightarrow i''\) in I, then the natural map \(l^* \circ f^* \rightarrow (f \circ l)^*\) is an isomorphism.
Remark 7.13
If I is an ordered category, so that L consists of identity maps, then every prefibration \(\mathcal {C}\rightarrow I\) is good.
Remark 7.14
If \(\mathcal {C}\rightarrow I\) is a Grothendieck fibration, then the second condition of Definition 7.12 is automatic.
Definition 7.15
 (i)
for any object \(i \in I\), C, F and W turn the fiber \(\mathcal {C}_i\) into a model category, and
 (ii)
for any morphism \(f:i' \rightarrow i\) in I, the transition functor \(f^*:\mathcal {C}_i \rightarrow \mathcal {C}_{i'}\) is rightderivable in the sense of Definition 3.1.
Definition 7.16
A functor \(\gamma :I' \rightarrow I\) between Reedy categories \(\langle I,L,M \rangle \), \(\langle I',L',M'\rangle \) is a Reedy functor if it sends \(L'\) to L and \(M'\) to M.
Theorem 7.17
 (i)
the restriction \(\mu ^*\mathcal {C}\rightarrow I_M\) is a Grothendieck fibration whose transition functors preserves finite limits,
 (ii)
for any object \(i \in I\), the matching category M(i) has an initial object.
We note for any Reedy category I, \((I_M)_L=(I_L)_M\) is discrete, so that as in Theorem 7.10, the model structure with the properties listed in Theorem 7.17 is unique. For any fixed model category \(\mathcal {C}\) and Reedy category I, \(\mathcal {C}\times I \rightarrow I\) is trivially a Reedy model prefibration, and \({\text {Sec}}(I,\mathcal {C}\times I) \cong \mathcal {C}^I\), so that Theorem 7.17 generalizes Theorem 7.10. The simplest new case of Theorem 7.17 appears for \(I=[1]\), and this is exactly the case that we have considered in Proposition 3.8.
To prove Theorem 7.17, Balzin generalizes the inductive description of \(\mathcal {C}^I\) given in Sect. 7.2. Namely, for any integer n, denote by \(\tau _{\le n}:I_{\le n} \rightarrow I\), \(\overline{\tau }_n:\overline{I}_n \rightarrow I\) the embedding functors, and for any good prefibration \(\mathcal {C}\rightarrow I\), consider the restriction functor \(U:{\text {Sec}}(I_{\le n},\tau _{\le n}^*\mathcal {C}) \rightarrow {\text {Sec}}(I_{\le n1},\tau _{\le n1}^*\mathcal {C})\). Then Balzin proves the following generalization of the equivalence (7.4).
Lemma 7.18
In the case (i) of Theorem 7.17, the same argument works, and since the functors \(l_!\) preserve colimits by adjunction, it also works for \(L_n\) in the general case. For \(M_n\), there is a problem: a fibration in \({\text {Sec}}(I_{\le n1},\mathcal {C})\) in general only restricts to a pointwise fibration in \(\mathcal {C}_i^{M(i)}\) and not necessarily to a fibration with respect to the injective model structure. However, under the assumption (ii) of Theorem 7.17, taking \(\mathsf{lim}_{M(i)}\) amounts to evaluating at the initial object of the category M(i), so that pointwise fibrations are good enough.
Remark 7.19
In fact, what Balzin proves in [1] is even more general than Theorem 7.17: instead of our notion of a good prefibration, he introduces a more general notion of a semifibration over a category with a factorization system. This treats both sides of the factorization system on an equal footing, and in the Reedy case, contains exactly enough structure to define the functors (7.7) and the map (7.3). We will not need this larger generality.
7.4 Matching expansions
We note that case (i) of Theorem 7.17 is only very slightly stronger than known results of Simpson and Hirschowitz [25]. The real novelty is in the case (ii), and at a first glance, the condition imposed on the Reedy category I is extremely restrictive. However, it turns out that it can always be satisfied by enlarging I.
Definition 7.20
The matching expansion M(I) of the Reedy category I is the full subcategory \(M(I) \subset (\Delta ^M_L I)^\perp \) spanned by nondegenerate objects.
Lemma 7.21
The functor \(\rho \) of (7.12) is a prefibration.
Proof
 (i)
\(f \in L\) iff \(\pi (f)^o\) is a special map in \(\Delta \) in the sense of Definition 5.1,
 (ii)
\(f \in M\) iff \(\pi (f)^o\) is cospecial and f is cocartesian with respect to \(\pi \).
Example 7.22
Let \(I=[1]\), with the degree function \(\deg (0)=1\), \(\deg (1)=0\). Then \(M(I) \cong \mathsf{V}\), with the degree function \(\deg (1)=0\), \(\deg (0)=1\), \(\deg (2)=2\) (as in Example 7.11).
Proposition 7.23
The matching expansion M(I) of a Reedy category I with the classes L, M and the degree function as above is a Reedy category, and it satisfies the condition (ii) of Theorem 7.17.
Proof
Next, observe that since the degree function on I is by definition nonnegative, we have \(\deg (i_0) \ge n\) for any nondegenerate diagram (5.3), so that the degree function (7.14) is also nonnegative. Moreover, the degree function (7.14) is chosen in such a way that the induced order on objects \(c \in M(I)\) is lexicographic: \(\deg (c) < \deg (c')\) iff either \(\deg (\rho (c)) < \deg (\rho (c'))\), or \(\deg (\rho (c)) = \deg (\rho (c'))\) and \(\deg (\pi (c')) < \deg (\pi (c))\).
For any object \(c = \langle [n],i_\cdot \rangle \in M(I)\), the matching category M(c) by definition consists of cocartesian liftings \(c \rightarrow f^o_!c\) of the maps \(f^o\) opposite to nonidentity cospecial maps \(f:[n'] \rightarrow [n]\). Thus it is equivalent to the category \([n1]\). In particular, it is finite and satisfies (ii) of Theorem 7.17. Moreover, for any such map \(f:[n'] \rightarrow [n]\), we have \(\deg (\rho (f^o_!c)) = \deg (i_{f(0)}) < \deg (i_0) = \deg (\rho (c))\), so that the cocartesian lifting of \(f^o\) lowers the degree.
For the latching category, observe that in fact a map \(f:c \rightarrow c'\) in M(I) is in the class L if and only if \(\rho (f)\) is the class L. Then if \(\rho (f)\) is not an identity map, we have \(\deg (\rho (c)) < \deg (\rho (c'))\), and if \(\rho (f)\) is an identity map but f is not, then \(\deg (\rho (c))=\deg (\rho (c'))\) but \(\deg (\pi (c)) > \deg (\pi (c'))\), so that in any case \(\deg (c) < \deg (c')\). Moreover, the latching category L(c) is the total space of the prefibration \(\rho :L(c) \rightarrow L(\rho (c))\) with fibers (7.13), the fibers are finite, and \(L(\rho (c))\) is also finite by assumption. Therefore L(c) is finite, and this finishes the proof. \(\square \)
7.5 Derived sections
One can ask whether the same holds globally over I. More precisely, let us introduce the following.
Definition 7.24
A section \(\sigma \in {\text {Sec}}(I,\mathcal {C})\) of a model prefibration \(\mathcal {C}\) over a small category \(\mathcal {C}\) is homotopy cartesian along a map f if the corresponding map (4.6) is a weak equivalence.
Definition 7.25
For any good model prefibration \(\mathcal {C}\) over a Reedy category I with the matching expansion \(\rho :M(I) \rightarrow I\), a derived section \(\sigma \) of \(\mathcal {C}\) over I is a section \(\sigma \in {\text {Sec}}(M(I),\rho ^*\mathcal {C})\) that is homotopy cartesian along all maps f in M(I) vertical with respect to \(\rho \). The full subcategory in \({\text {Sec}}(M(I),\rho ^*\mathcal {C})\) spanned by derived sections is denoted by \({\text {Sec}}_{\rho }(M(I),\rho ^*\mathcal {C})\).
Unfortunately, we do not know whether this statement is always true. So, for a general good model prefibration \(\mathcal {C}/I\), we simply take \({\text {DSec}}(I,\mathcal {C})\) as a correct replacement for \({\text {Ho}}({\text {Sec}}(I,\mathcal {C}))\). As a justification for this, let us prove that the two agree in the cases covered by Theorem 7.17,
Proposition 7.26
Assume given a good model prefibration \(\mathcal {C}\) over a Reedy category I that satisfies either of the two conditions of Theorem 7.17. Then the functor (7.19) is an equivalence of categories.
Proof
In the case (ii) of Theorem 7.17, note that by induction, the matching category M(i) for any \(i \in I\) is an ordinal category [n]. Then the fiber \(M(I)_i\) of the prefibration \(\rho \) has an initial object corresponding to the longest nondegenerate diagram (5.3). Sending i to this initial object defines a section \(\sigma :I \rightarrow M(I)\) of the prefibration \(\rho \), and \(\sigma \) is leftadjoint to \(\rho \). Therefore \(\sigma ^*\) is rightadjoint to the fully faithful functor (7.16). Since both functors \(\sigma ^*\) and \(\rho ^*\) obviously preserve weak equivalences, they induce an adjoint pair of functors on homotopy categories, and since \(\rho \circ \sigma = \mathsf{id}\), the functor \({\text {Ho}}(\rho ^*):{\text {Ho}}({\text {Sec}}(I,\mathcal {C})) \rightarrow {\text {Ho}}({\text {Sec}}(M(I),\rho ^*\mathcal {C}))\) induced by (7.16) is fully faithful. Moreover, for any derived section s of \(\mathcal {C}\) over I, the adjunction map \(\rho ^*\sigma ^*s \rightarrow s\) is a pointwise weak equivalence, so that it lies in the image of the fully faithful functor \({\text {Ho}}(\rho ^*)\).
To do this, first note that with the new Reedy structure on M(I), we have \(M(I)_M=M(I_M)\), so by Theorem 7.17, we can replace I with \(I_M\) and assume right away that I is an ordered category (and \(\mathcal {C}\) is then a Grothendieck fibration). Moreover, for any discrete Grothendieck cofibration \(\kappa :I' \rightarrow I\), \(I'\) is also an ordered category, and \(M(\kappa ):M(I') \rightarrow M(I)\) is a discrete Grothendieck cofibration that induces equivalences of matching categories. Then \(M(\kappa )^*\) commutes with matching functors, thus is rightderivable. Therefore we can replace I with the category I(i) and assume that \(i \in I\) is the initial object. For the final reduction, note that we then have the constant functor \(s:I \rightarrow I\) with value \(i \in I\) and the map \(a:s \rightarrow \mathsf{id}\), and a induces a functor \(a^*:\mathcal {C}\rightarrow \mathcal {C}_i \times I\) to the constant Grothendieck fibration with fiber \(\mathcal {C}_i\). Moreover, this functor is cartesian, and for any \(i' \in I\), \(a(i')\) commutes with finite limits. Then by (7.9), \(a^*\) commutes with matching functors, thus is rightderivable, so that we may assume that \(\mathcal {C}\rightarrow I\) is a constant Grothendieck fibration with fiber \(\mathcal {C}_i\), and \({\text {Sec}}(M(I),\rho ^*\mathcal {C})\) is then the functor category \(\mathcal {C}_i^{M(I)}\) with the injective model structure. But since \(\rho :M(I) \rightarrow I\) is a prefibration, the embedding \(\varepsilon (i):M(I)_i \rightarrow M(I)\) of the fiber over the initial object i admits a rightadjoint functor \(\varepsilon (i)^\dagger \), and the pullback functor \((\varepsilon (i)^\dagger )^*\) is then leftadjoint to \(\varepsilon (i)^*\). Since any pullback functor is trivially leftderivable for injective model structures, \(\varepsilon (i)^*\) is rightderivable by adjunction. \(\square \)
8 Reedy categories II
8.1 Derived sections of stable pairs
In the situation of stable model pairs, it is easy to show that Balzin model structure of Theorem 7.17 also gives a stable model pair (this also works in the classical Reedy case).
Definition 8.1
A stable model prefibration \(\langle \mathcal {C},\mathcal {C}' \rangle \) over a category I is a model prefibration \(\mathcal {C}' \rightarrow I\) equipped with a full subcategory \(\mathcal {C}\subset \mathcal {C}'\) such that for any \(i \in I\), \(\langle \mathcal {C}_i,\mathcal {C}'_i \rangle \) is a stable model pair, and for any morphism \(f:i' \rightarrow i\) in I, the transition functor \(f^*:\mathcal {C}'_i \rightarrow \mathcal {C}'_{i'}\) is a stable rightderivable functor between stable model pairs. A stable model prefibration \(\langle \mathcal {C},\mathcal {C}' \rangle \) over a Reedy category I is good if \(\mathcal {C}' \rightarrow I\) is good in the sense of Definition 7.12.
Proposition 8.2
Proof
For \(L_n\), this is clear: the functors \(l_!\) in (7.7) are leftQuillen by Definition 7.12, so that \(L_n\) is also leftQuillen, and for any \(i \in M(I)\), \(\mathcal {C}_i \subset \mathcal {C}'_i\) is closed under homotopy colimits by Definition 2.12 (ii), \(L^\cdot L_n\) sends sections of \(\rho ^*\mathcal {C}\subset \rho ^*\mathcal {C}'\) into sections of \(\rho ^*\mathcal {C}\).
For \(M_n\), recall that the Reedy category M(I) satisfies the assumption (ii) of Theorem 7.17, so that taking the limit in (7.7) amounts to evaluating at the initial object of the category M(i). Proposition 3.12 immediately implies by induction that homotopy cartesian squares in \({\text {Sec}}(M(I)_{\le n},\tau ^*_{\le n}\rho ^*\mathcal {C}')\) are pointwise homotopy cartesian, and since the transition functors \(f^*\) in (7.7) are stable by Definition 8.1, the matching functor \(M_n\) is also stable.
But now, by virtue of (8.2), \({\text {Sec}}_\rho (M(I),\rho ^*\mathcal {C}) \subset {\text {Sec}}(M(I),\rho ^*\mathcal {C}')\) satisfies Definition 2.12 (i) automatically, so that it remains to check Definition 2.12 (ii). This is obvious: by Proposition 3.12, a square in the category \({\text {Sec}}(M(I),\rho ^*\mathcal {C})\) is homotopy cartesian or cocartesian if and only if this holds pointwise, and the transition functor \(f^*\) for any map f in M(I) vertical with respect to \(\rho \) is an equivalence of categories, thus sends homotopy cartesian squares to homotopy cartesian squares. \(\square \)
Lemma 8.3
A triangle in the category \({\text {DSec}}(I,\mathcal {C})\) is distinguished if and only if it becomes distinguished after evaluating at any object \(i \in I\), and for any Reedy functor \(\gamma :I' \rightarrow I\), the pullback functor (8.3) is triangulated.
Proof
Since the model structure on \({\text {Sec}}(M(I),\mathcal {C}')\) is constructed by iterated application of Proposition 3.12, a square in \({\text {Sec}}(M(I),\mathcal {C}')\) is homotopy cartesian resp. cocartesian iff this holds pointwise. Thus a triangle in \({\text {Ho}}({\text {Sec}}(M(I),\rho ^*\mathcal {C}))\) is distinguished iff it becomes distinguished after evaluation as any object \(i \in M(I)\). Since the class of distinguished triangles is stable under weak equivalence, for triangles in \({\text {DSec}}(I,\mathcal {C}) \subset {\text {Ho}}({\text {Sec}}(M(I),\rho ^*\mathcal {C}))\), it suffices to evaluate at the terminal objects of the fibers \(M(I)_i\) of the prefibration \(\rho :M(I) \rightarrow I\). This proves the first claim; the second one is an immediate corollary of the first. \(\square \)
8.2 Filtration by degree
Proposition 8.4
Lemma 8.5
Proposition 8.4 holds if the model prefibration \(\mathcal {C}' \rightarrow I\) satisfies the assumptions of Theorem 7.17.
Proof
By Proposition 7.26, in the case under consideration, \({\text {DSec}}(I,\mathcal {C})\) is equivalent to the homotopy category \({\text {Ho}}({\text {Sec}}(I,\mathcal {C})) \subset {\text {Ho}}({\text {Sec}}(I,\mathcal {C}'))\). On the level of model categories, the adjoint functors \(U^\dagger \) and \(U_\dagger \) are then provided by Lemma 7.17, and as we saw in the proof of Proposition 8.2, both are derivable and stable. By Lemma 7.18, the statement then reduces to Lemma 3.16. \(\square \)
Moreover, let \(\overline{M(I)}_n = \rho ^{1}(\overline{I}_n) \subset M(I_{\le n}) \subset M(I)\) be the union of the fibers \(M(I)_i\) of the prefibration \(\rho :M(I) \rightarrow I\) over all objects \(i \in I\) of degree \(\deg (i)=n\), and denote by \(\varepsilon _n:\overline{M(I)}_n \subset M(I_{\le n})\) the embedding functors. Note that \(\overline{M(I)}_n\) is a Reedy category, in fact the opposite to an ordered category, and \(\langle \varepsilon _n^*\rho ^*\mathcal {C},\varepsilon _n^*\rho ^*\mathcal {C}' \rangle \) is a good stable model prefibration over \(\overline{M(I)}_n\) satisfying either of the conditions of Theorem 7.17. On a component \(M(I)_i \subset \overline{M(I)}_n\) corresponding to an object \(i \in \overline{I}_n\), it is actually constant with fiber \(\langle \mathcal {C}_i,\mathcal {C}'_i \rangle \).
Lemma 8.6
Proof
Let us prove it for all \(l \ge 2(n1)^2\). Assume by induction that the claim is already proved for \(l1\), and note that since M(I) satisfies the assumptions of Theorem 7.17 (ii) by Proposition 7.23, and \(M(I)_L\) satisfies this assumption trivially, we are within the scope of Lemma 8.5. Then in particular, we know that \(\lambda ^*\) induces an equivalence \({\text {DSec}}_l(M(I)_L,\rho ^*\mathcal {C}) \cong {\text {DSec}}_l(M(I),\rho ^*\mathcal {C})\), and moreover, since it is leftderivable, it commutes with the adjoint functor \(U_\dagger \) by (7.8). Then to finish the proof, it suffices to show that \(\lambda ^*\) also commutes with \(U^\dagger \). Since for any map \(m:i \rightarrow i'\) in \(M(I)_M\) with \(2(n1)^2 < \deg (i) \le 2n^2\), we have \(\deg (i') \le 2(n1)^2\), this trivially follows from (7.7). \(\square \)
Proof of Proposition 8.4
8.3 Deriving latching and matching functors
In order to apply Proposition 8.4, it is useful to obtain a version of the isomorphisms (7.7) and (7.8) valid for derived sections. We start with (7.8).
Lemma 8.7
The maps (8.15) are isomorphisms.
Proof
Since functors \(U^\dagger \), \(V^\dagger \) and \(U_\dagger \), \(V_\dagger \) are the projection functors in a semiorthogonal decomposition, it suffices to prove that only one of the maps in each line is an isomorphism—the other one then will also be an isomorphism.
For \(\lambda \), we have \(M(I)_L \cong \lambda ^*M(I)\), and if we denote by \(\widetilde{\lambda }:M(I)_L \rightarrow M(I)\) the embedding, then the same argument shows that \(\widetilde{V}_\dagger \circ \widetilde{\lambda }^* \cong \widetilde{\lambda }^* \circ \widetilde{V}_\dagger \). But now \(M(I_L) \cong I_L\), and the embedding \(M(\lambda ):I_L = M(I_L) \rightarrow M(I)_L \subset M(I)\) is rightadjoint to the projection \(\rho :M(I)_L = \lambda ^*M(I) \rightarrow I_L\), so that \(M(\lambda )^*\) is exactly the leftadjoint functor to the embedding (8.16). Therefore by definition, \(V_\dagger \circ M(\lambda )^* \cong M(\lambda )^* \circ \widetilde{V}_\dagger \), so that altogether, \(V_\dagger \circ \lambda ^* \cong \lambda ^* V_\dagger \) and \(U_\dagger \circ \lambda ^* \cong \lambda ^* \circ U_\dagger \). \(\square \)
Definition 8.8
A full subcategory \(I' \subset I\) in a Reedy category I is leftclosed if for any map \(l:i' \rightarrow i\) in L with \(i \in I'\), \(i'\) is also in \(I'\).
Note that for any map \(f:i \rightarrow i'\) in \(I'\) with the canonical decomposition \(f = l \circ m\), \(m:i \rightarrow i''\) in M, \(l:i'' \rightarrow i'\) in L, Definition 8.8 implies that \(i''\), hence the whole decomposition also lies in \(I'\). Thus \(I'\) with the induced degree functors and classes L, M is aso a Reedy category, and the embedding \(\gamma :I' \rightarrow I\) is a degreepreserving Reedy functor. If \(I = I_M\) is an ordered subcategory, then any full subcategory \(I' \subset I\) is trivially leftclosed; in particular, for any leftclosed \(I' \subset I\), \(I'_M \subset I_M\) is leftclosed.
Lemma 8.9
Assume given a good stable model prefibration \(\langle \mathcal {C},\mathcal {C}' \rangle \) over a Reedy category I, and a leftclosed full subcategory \(I' \subset I\) with the embedding functor \(\gamma :I' \rightarrow I\). Then the pullback functor \(\gamma ^*\) of (8.3) admits a fully faithful rightadjoint functor \(\gamma _*:{\text {DSec}}(I',\gamma ^*\mathcal {C}) \rightarrow {\text {DSec}}(I,\mathcal {C})\). Moreover, if we denote by \(\gamma ^M:I_M \rightarrow I'_M\) the induced embedding, then the base change map \(\mu ^* \circ \gamma _* \rightarrow \gamma ^M_* \circ \mu ^*\) is an isomorphism.
Proof
By definition, we have to show that for any \(E \in {\text {DSec}}(I',\gamma ^*\mathcal {C})\), the functor \(E' \mapsto {\text {Hom}}(\gamma ^*E',E)\) from \({\text {DSec}}(I,\mathcal {C})\) to abelian groups is represented by a derived section \(\gamma _*(E) \in {\text {DSec}}(I,\mathcal {C})\), and the natural map \(\gamma ^*\gamma _*E \rightarrow E\) is an isomorphism. Assume first that \(I = I_{\le m}\) for some integer \(m \ge 0\). Then Proposition 8.4 implies by induction that for any n, the restriction functors \(\tau _{\le n}^*\) for the derived sections of the prefibrations \(\mathcal {C}\) and \(\gamma ^*\mathcal {C}\) admit rightadjoint fully faithful triangulated functors \(\tau ^{\le n}_*\), and every derived section E of \(\gamma ^*\mathcal {C}\) is a finite extension of derived sections of the form \(\tau ^{\le n}_*E_n\), \(E_n \in {\text {DSec}}_n(I'_{\ge n},\gamma ^*\mathcal {C})\). Thus it suffices to construct \(\gamma _*(E)\) for E of this form. But by definition, we have \(\tau _{\le n}^* \circ \gamma ^* \cong \gamma ^* \circ \tau _{\le n}^*\), and the functor \(\gamma ^*:{\text {DSec}}_n(I,\mathcal {C}) \rightarrow {\text {DSec}}_n(I',\gamma ^*\mathcal {C})\) is a projection onto a direct summand, thus trivially has a fully faithful rightadjoint functor \(\gamma _*\). Moreover, by (8.18) and Definition 8.8, the restriction functor \(\gamma ^*:{\text {DSec}}_{\le n}(I,\mathcal {C}) \rightarrow {\text {DSec}}_{\le n}(I',\gamma ^*\mathcal {C})\) commutes with the functor \(U_\dagger \), so that \(\gamma _*E_n \in {\text {DSec}}_n(I',\gamma ^*\mathcal {C}') \subset {\text {DSec}}_{\le n}(I,\mathcal {C})\) also represents the desired functor. Then letting \(\gamma _*(\tau ^{\le n}_*E_n)) = \tau ^{\le n}_*\gamma _*E_n \in {\text {DSec}}(I,\mathcal {C})\) also gives the correct representing object, and Lemma 8.7 then shows that \(\mu ^* \circ \gamma _* \cong \gamma ^M_* \circ \mu ^*\).
Lemma 8.10
Proof
The fact that \(I'' \subset I\) is leftclosed is obvious (for any map \(f:i \rightarrow i'\) whatsoever with \(i \in I''\), we cannot have \(i' \in I'\)). For the second claim, since \(\gamma '_*\) is fully faithful, it suffices to prove that \(\gamma ^*(\gamma _*'(E))=0\) for any \(E \in {\text {DSec}}(I'',\gamma ^{'*}\mathcal {C})\), and as in the proof of Lemma 8.9, it suffice to do it for \(E = \tau ^{\le n}_*E_n\), \(E_n \in {\text {DSec}}_n(I'',\gamma ^{'*}\mathcal {C})\). But it immediately follows from our additional assumption on \(I' \subset I\) that the embedding \(\gamma \) induced an equivalence \(M_i \rightarrow M_{\gamma (i)}\) of matching categories for any \(i \in I'\), so that by (8.25), \(\gamma ^*\) commutes with the functors \(\tau ^{\le n}_*\). Thus it suffices to consider the case \(I=I_{\le n}\), \(\gamma '_*(E)=\gamma '_*(E_n) \in {\text {DSec}}_n(I,\mathcal {C})\), where the claim is obvious. The existence of the adjoint functor \(\gamma _!\) then immediately follows from the existence of the semiorthogonal decomposition. However, it can also be constructed explicitly by induction on degree, as in Lemma 8.9, and then as in Lemma 8.9, the isomorphism \(\gamma ^L_! \circ \lambda ^* \rightarrow \lambda ^* \circ \gamma _!\) immediately follows from Lemma 8.7. \(\square \)
8.4 Fundamental spectral sequence
Corollary 8.11
Here \({\text {Hom}}^\cdot (,)\) is understood in the standard triangulated category sense: \({\text {Hom}}^m(A,B) = {\text {Hom}}(A,B[m])\) for any integer m, where [m] is the homological shift with respect to the triangulated structure provided by Proposition 2.20.
Proof
Example 8.12
To use effectively the spectral sequence of Corollary 8.11, we need some way to control the functors \(\overline{L}_i\), \(\overline{M}_i\), \(i \in I\). It is convenient to introduce the following.
Definition 8.13
For any object i in a Reedy category I, the completed latching category \(\overline{L}(i)\) is the category of objects \(i' \in I\) equipped with a map \(l:i' \rightarrow i\) in the class L, and the completed matching category \(\overline{M}(i)\) is the category of objects \(i' \in I\) equipped with a map \(m:i \rightarrow i'\) in the class M.
Now, for any finite ordered category J with an initial object \(o \in J\), and any pointed model category \(\mathcal {C}\), evaluation at o has a rightadjoint functor \(S:{\text {Ho}}(\mathcal {C}) \rightarrow {\text {Ho}}(\mathcal {C}^J)\) sending \(E \in \mathcal {C}\) to a functor \(E':J \rightarrow \mathcal {C}\) such that \(E'(o)=E\) and \(E'(j)=0\) for \(j \ne o\), and the functor S in turn has a rightadjoint functor \(S^\dagger :{\text {Ho}}(\mathcal {C}^J) \rightarrow {\text {Ho}}(\mathcal {C})\). For a stable model pair \(\langle \mathcal {C},\mathcal {C}' \rangle \), this induces a triangulated functor \(S^\dagger :{\text {Ho}}(\mathcal {C}^J) \rightarrow {\text {Ho}}(\mathcal {C})\). Dually, we also have a functor \(T:{\text {Ho}}(\mathcal {C}) \rightarrow {\text {Ho}}(\mathcal {C}^{J^o})\) leftadjoint to the evaluation at o, and it has a leftadjoint functor \(T_\dagger :{\text {Ho}}(\mathcal {C}^{J^o}) \rightarrow {\text {Ho}}(\mathcal {C})\) and its triangulated version for stable model pairs. If \(J=[1]\), then these are exactly the functors (2.6), but they also exist for a more general category J. In particular, we can take \(J=\overline{M}(i)\) or \(J=\overline{L}(i)^o\), with the initial object o given by the identity arrow \(\mathsf{id}:i \rightarrow i\) in both cases.
Lemma 8.14
Proof
Immediately follows from (8.18) resp. (8.25) applied both to I and to \(\overline{L}(i)\) resp. \(\overline{M}(i)\). \(\square \)
Corollary 8.15
Proof
We obviously have \(\widetilde{\lambda }(i) \cong \widetilde{\lambda }(o) \circ \lambda (i)\) and \(\widetilde{\mu }(i) \cong \widetilde{\mu }(o) \circ \mu (i)\). \(\square \)
For another useful corollary of Lemma 8.14, assume given a good stable model prefibration \(\langle \mathcal {C},\mathcal {C}' \rangle \) over a Reedy category I, and a leftclosed full subcategory \(I' \subset I\) with the embedding functor \(\gamma :I' \rightarrow I\), as in Lemma 8.9. Moreover, assume that the full embedding \(\gamma :I' \rightarrow I\) admits a leftadjoint functor \(\gamma _\dagger :I \rightarrow I'\), with the adjunction map \(a:\mathsf{Id}\rightarrow \gamma \circ \gamma _\dagger \) compatible with the prefibration \(\mathcal {C}' \rightarrow I\) in the sense of Definition 4.17.
Corollary 8.16
In the assumptions above, the essential image of the fully faithful functor \(\gamma _*:{\text {DSec}}(I',\gamma ^*\mathcal {C}) \rightarrow {\text {DSec}}(I,\mathcal {C})\) consists of derived sections that are homotopy cartesian along \(a(i):i \rightarrow \gamma (\gamma _\dagger (i))\) for all \(i \in I\).
Proof
8.5 Functoriality
Another corollary of Proposition 8.4 is extended functoriality for the categories of derived sections. Namely, assume given a Reedy category I and two good stable model prefibrations \(\langle \mathcal {C}_0,\mathcal {C}_0' \rangle \), \(\langle \mathcal {C}_1,\mathcal {C}_1' \rangle \) over I.
Definition 8.17
A stable morphism \(\Phi \) from \(\langle \mathcal {C}_0,\mathcal {C}_0' \rangle \) to \(\langle \mathcal {C}_1,\mathcal {C}_1' \rangle \) is a functor \(\Phi :\mathcal {C}_0' \rightarrow \mathcal {C}_1'\) over I such that for any object \(i \in I\), \(\Phi (i)\) is a stable rightderivable functor in the sense of Definition 3.2.
Lemma 8.18
The pair \(\langle \widetilde{\mathcal {C}},\widetilde{\mathcal {C}}' \rangle \) is a good stable model prefibration over the Reedy category \(I \times [1]^o\).
Proof
Lemma 8.19
The map (8.42) is compatible with the differentials, and the \(E_1\)term \(E_1(A,B)\) of the spectral sequence of Corollary 8.11 with its differential is the cone of this map.
Proof
9 General gluing
We can now combine the homotopical results of Sects. 7 and 8, on one hand, and the simplicial technology of Sects. 5 and 6, on the other hand. The result is a derived version of the gluing constructions of Sect. 4.5. In this section, we treat the case of cosections, with comonads and coalgebras postponed until Sect. 10. We define the notion of a stable model family over a category I, our version of a family of enhanced triangulated categories, and for any such family \(\mathcal {C}\), we construct the triangulated category \({\text {DRec}}(I,\mathcal {C})\) of derived cosections. We show that \({\text {DRec}}(I,\mathcal {C})\) has the same basic functoriality as the usual category of cosections of Definition 4.21. We then construct a spectral sequence that refines the isomorphism of Lemma 5.13 and gives one some control over \({\text {DRec}}(I,\mathcal {C})\). As an application, we show that a triangulated category coming from a stable model pair is automatically enriched over the stable homotopy category, and prove additional results on functoriality in the case of a constant family \(\mathcal {C}= \mathcal {C}_0 \times I\), when \({\text {DRec}}(I,\mathcal {C})\) is the enhanced version of the category of functors from I to \({\text {Ho}}(\mathcal {C}_0)\). In particular, we construct the left and rightderived Kan extension functors.
9.1 Stable model families
Let us apply the technology of Reedy model structures to special prefibrations over categories of simplices in the sense of Definition 5.1.
Recall that for any simplicial set X, the category of simplices \(\Delta X\) carries a natural Reedy structure of Example 7.8. The opposite category \((\Delta X)^o\) is then also a Reedy category, with the opposite Reedy structure.
Lemma 9.1
A prefibration \(\mathcal {C}\rightarrow (\Delta X)^o\) special in the sense of Definition 5.10 is good in the sense of Definition 7.12 if and only if it is normalized.
Proof
Every surjective map in \(\Delta \) is special, so that Definition 7.12 (i) follows from Definition 5.10, and Definition 7.12 (ii) is then precisely equivalent to the normalization condition. \(\square \)
Definition 9.2
For any simplicial set X, a good stable model prefibration \(\langle \mathcal {C},\mathcal {C}' \rangle \) over the category \((\Delta X)^o\) is homotopy special if the corresponding prefibration (8.1) is special in the sense of Definition 5.10. A derived section \(E \in {\text {DSec}}((\Delta X)^o,\mathcal {C})\) of such a stable model prefibration is homotopy special if the section \(\mathsf{ev}(E)\) obtained by applying the evaluation functor (8.5) is special in the sense of Definition 5.10.
An abundant source of normalized special prefibrations over categories of simplices is provided by Proposition 6.11. To incorporate model structures into the picture, we introduce the following.
Definition 9.3
A model family \(\mathcal {C}\) over a small category I is a category \(\mathcal {C}\) equipped with a precofibration \(\mathcal {C}\rightarrow I\) and classes of fiberwise maps C, F, W such that for every \(i \in I\), the fiber \(\mathcal {C}_i\) with the classes C, F, W is a model category, and for every map \(f:i \rightarrow i'\) in I, the transition functor \(f_!:\mathcal {C}_i \rightarrow \mathcal {C}_{i'}\) is rightderivable. A stable model family over I is a model family \(\mathcal {C}' \rightarrow I\) equipped with a full subcategory \(\mathcal {C}\subset \mathcal {C}'\) such that for any \(i \in I\), \(\langle \mathcal {C}_i,\mathcal {C}'_i \rangle \) is a stable model pair, and for any map \(f:i \rightarrow i'\), the functor \(f_!\) is stable.
Then for any model family \(\mathcal {C}\rightarrow I\), the barycentric expansion \(S(I,\mathcal {C}) \rightarrow \Delta I\) of Definition 6.9 is a model prefibration in the sense of Definition 7.15. Moreover, by Lemma 9.1, its reflection \(S(I,\mathcal {C})^\flat \) of Definition 5.12 is good in the sense of Definition 7.12. Recall that explicitly, the fiber \(S(I,\mathcal {C})^\flat _{\langle [n],\kappa \rangle }\) over an object \(\langle [n],\kappa \rangle \in (\Delta I)^o\) is naturally identified with the fiber \(\mathcal {C}_{\kappa (n)}\). For a stable model family \(\langle \mathcal {C},\mathcal {C}' \rangle \), we have a good model prefibration \(\widetilde{\mathcal {C}'} = S(I,\mathcal {C}')^\flat \rightarrow (\Delta I)^o\), and if we let \(\widetilde{\mathcal {C}} \subset \widetilde{\mathcal {C}}'\) be the full subcategory spanned by the subcategories \(\mathcal {C}_{\kappa (n)} \subset \mathcal {C}'_{\kappa (n)} \cong \widetilde{\mathcal {C}}'_{\langle [n],\kappa \rangle }\), then \(\langle \widetilde{\mathcal {C}},\widetilde{\mathcal {C}'} \rangle \) is obviously a good stable model prefibration over \(\Delta I\) homotopy special in the sense of Definition 9.2.
Definition 9.4
Lemma 9.5
Proof
9.2 The standard differential
Let us now show that for homotopy special sections A, B of homotopy special good stable model prefibrations, one can actually compute the first differential and the \(E_2\)term of the spectral sequence of Corollary 8.11. Recall that a simplex \(x \in X([n])\) in a simplicial set X is nondegenerate if it does not lie in the image of the map \(X(f):X([n']) \rightarrow X([n])\) for a nonidentical surjective map \(f:[n] \rightarrow [n']\).
Lemma 9.6
Proof
For any simplex \(x \in X([n])\), there exists a unique nondegenerate simplex \(x' \in X([n'])\) and surjective map \(f:[n] \rightarrow [n']\) such that \(x = f^*x'\). The completed latching category \(\overline{L}(\langle [n],x \rangle )\) of Definition 8.13 is then naturally equivalent to the product \([1]^{nn'}\) of \(nn'\) copies of the category [1]. If x itself is nondegenerate, this is just the point, and then Lemma 8.14 immediately provides the isomorphism (9.8). If not, then by Lemma 8.14, \(\overline{L}_{\langle [n],x \rangle }\) can still be computed after restriction to \(\overline{L}(\langle [n],x \rangle ) \cong [1]^{nn'}\), and the corresponding functor \(\overline{L}_o\) on the completed latching category is obtained by applying the functor \(T_\dagger \) of (2.6) along each of the \(nn'\) coordinates. Since E is homotopy special, \(\lambda (\langle [n],x \rangle )^*E\) sends any map in \(\overline{L}(\langle [n],x \rangle )\) to a weak equivalence, and we obtain 0 already after the first application of \(T_\dagger \). \(\square \)
Proposition 9.7
Proof
The construction is obviously functorial in X and B, so to finish the proof, it remains to check that (9.14) is compatible with the differentials. However, since the spectral sequence of Corollary 8.11 is functorial with respect to base change, this can also be checked after restricting to \(\overline{M}(\langle [n],x \rangle )\). Then it immediately follows from Lemma 8.19 by induction on n. \(\square \)
Lemma 9.8
For any \(A,B \in {\text {Ho}}(\mathcal {C})\), the functors (9.17) together with the maps (9.18) form a generalized cohomology theory.
Proof
All the axioms except for excision and homotopy invariance are obvious. For excision and homotopy invariance, use the spectral sequence of Corollary 8.11 to compute \(H^\cdot (A,B)(X,Y)\), and note that the functor \({\mathcal {H}{} { om}}(A_X,B_X)\) is actually constant for any simplicial set X, so that the righthand side of (9.11) reduces to the ordinary relative cohomology \(H^\cdot (X,Y;)\). Thus by Proposition 9.7, both properties hold already in the \(E_2\)term. \(\square \)
Remark 9.9
By Lemma 9.8, for any \(A,B \in {\text {Ho}}(\mathcal {C})\), we have a an object \(H^\cdot (A,B)\) in the stable homotopy category, welldefined up to an isomorphism.
Of course, this is much weaker than even the usual categorical notion of enrichment—proving that would require defining \(H^\cdot (A,B)\) up to a unique isomorphism and checking that the correspondence \(A,B \mapsto H^\cdot (A,B)\) is compatible with compositions, and it does not seem possible to do either in terms of generalized cohomology theories.
9.3 Specialization
 (i)
the projection \(\mathcal {C}' \rightarrow I\) is a Grothendieck cofibration, and
 (ii)
for any map \(f:i \rightarrow i'\) in I, the transition functor \(f_!:\mathcal {C}'_i \rightarrow \mathcal {C}'_{i'}\) preserves finite limits and sends \(\mathcal {C}_i \subset \mathcal {C}'_i\) into \(\mathcal {C}_{i'} \subset \mathcal {C}'_{i'}\).
As it turns out, if restrict our attention to homotopy special derived sections, then we do not even need to consider the simplicial expansion \(\Delta I\).
Proposition 9.10
In order to prove this, we need one preliminary result. Consider the right commacategory \(\mathsf{R}(\beta ^o)\) of the projection \(\beta ^o:(\Delta I)^o \rightarrow I^o\), with its projections (3.2), and for any object \(i \in I\), let \((\Delta I)^o_i = \tau _i\) be the fiber of the projection \(\tau :\mathsf{R}(\beta ^o) \rightarrow I^o\) over i. Note that \((\Delta I)^o_i \cong (\Delta I/i)^o\) is opposite to the simplicial replacement of the category I / i of objects \(i' \in I\) equipped with an arrow \(i' \rightarrow i\). Thus \((\Delta I)^o_i\) has a natural Reedy structure.
Lemma 9.11
Assume given a model category \(\mathcal {C}\), and equip the category of functors \(\mathcal {C}^{(\Delta I)^o_i}\) with the corresponding Reedy model structure. Then the tautological pullback functor \(\tau ^*:\mathcal {C}\rightarrow \mathcal {C}^{(\Delta I)^o_i}\) is leftderivable. Moreover, if \(\mathcal {C}\) has all limits, then \(L^\cdot \tau ^*\) is fully faithful and has a rightadjoint.
Proof
For any object \(\widetilde{i} = \langle [n],i_\cdot \rangle \in (\Delta I)^o_i\), the latching category \(L(\widetilde{i})\) is equivalent to the full subcategory \([1]^l \setminus 1^l \subset [1]^l\) for some l, and in particular, it has an initial object. Therefore for any constant functor \(c:(\Delta I)^o_i \rightarrow \mathcal {C}\), the natural map \(c(\widetilde{i}) \rightarrow L_n(c)(\widetilde{i})\) is an isomorphism, and \(\tau ^*\) is trivially leftderivable. Since \(\tau ^*\) also sends weak equivalences to weak equivalences, we have \({\text {Ho}}(\tau ^*) \cong L^\cdot \tau ^*\). If \(\mathcal {C}\) has all limits, then \(\tau ^*\) has a rightadjoint given by \(\mathsf{lim}_{(\Delta I)^o_i}\), so that \({\text {Ho}}(\tau ^*)\) has a rightadjoint by Quillen adjunction. Moreover, if we twist \(\Delta I\) by the involution \(\iota :\Delta \rightarrow \Delta \) of (5.12), then \((\Delta I)^o_i \cong (\Delta _< I)_i^o\), and we have the adjoint functors \(\rho ^o_i\), \(\lambda ^o_i\) of (6.9). On the other hand, the category \((\Delta _+ I)^o_i\) has a terminal object \(\langle [0],i \rangle \), so that the tautological functor \(\tau _+:(\Delta _+ I)^o_i\) has a rightadjoint functor \(\tau ^\dagger _+\). All the pullback functors \(\rho ^{o*}_i\), \(\lambda ^{o*}_i\), \(\tau _+^*\), \(\tau ^{\dagger *}_+\) send weak equivalences to weak equivalences, thus induce two pairs of adjoint functors between homotopy categories, and since \(\tau ^\dagger _+\) and \(\rho _i^o\) are fully faithful, \({\text {Ho}}(\tau ^*) \cong {\text {Ho}}(\lambda ^{o*}_i) \circ {\text {Ho}}(\tau _+^*)\) is also fully faithful. \(\square \)
Proof of Proposition 9.10
Remark 9.12
In Proposition 9.10, one does not need \(\mathcal {C}'\) to have all limits—it suffices to have limits of the size of the category \(\Delta I\) (e.g. countable if I is countable). However, as we shall see in Sect. 12.2, even the stronger condition is often harmless, since under a mild additional assumption, it can always be achieved by enlarging \(\mathcal {C}'\).
For example, for any stable model pair \(\langle \mathcal {C},\mathcal {C}' \rangle \) and small category I, the constant Grothendieck cofibration \(\mathcal {C}' \times I \rightarrow I\) tautologically satisfies the conditions (i), (ii) of Proposition 9.10, and if \(\mathcal {C}'\) has all limits, then we are within the scope of the Proposition, with all it entails (in particular, the relative category \(\langle \mathcal {C}^I,W^I \rangle \) is localizing, and the homotopy category \({\text {Ho}}(\mathcal {C}^I)\) is canonically triangulated). For model categories, this is the situation considered in [14], and all the results including Lemma 9.11 appear there. In this situation, Proposition 9.10 has a useful corollary (this is one of the main results of [14], but since we want it for model pairs, we reproduce the proof).
Proposition 9.13
Assume given a stable pair \(\langle \mathcal {C},\mathcal {C}' \rangle \), small categories \(I_0\), \(I_1\), and a functor \(\gamma :I_0 \rightarrow I_1\), and consider the corresponding pullback functor \(\gamma ^*:{\text {Ho}}(\mathcal {C}^{I_1}) \rightarrow {\text {Ho}}(\mathcal {C}^{I_0})\). Then if \(\mathcal {C}'\) has all colimits and \(\mathcal {C}\subset \mathcal {C}'\) is closed under them, the functor \(\gamma ^*\) has a leftadjoint functor \(\gamma _!\), and if \(C'\) has all limits and \(\mathcal {C}\subset \mathcal {C}'\) is closed under them, \(\gamma ^*\) has a rightadjoint functor \(\gamma _*\).
Proof
10 Comonads
10.1 Derived coalgebras
Assume given a stable model pair \(\langle \mathcal {C},\mathcal {C}' \rangle \). By a stable comonad on \(\langle \mathcal {C},\mathcal {C}' \rangle \) we will understand a comonad \(\Phi \) on \(\mathcal {C}'\) that is rightderivable as a functor from \(\mathcal {C}'\) to itself and stable in the sense of Definition 3.2. Given such a comonad, we can consider its barycentric expansion \(S(\mathcal {C}',\Phi )\) of Definition 6.2, but since it is not normalized in the sense of Definition 5.1, the reflection \(S(\mathcal {C}',\Phi )^\flat \) is not a good model prefibration over \(\Delta ^o\), and the methods of Sect. 7 do not apply.
The standard way to get around this problem is to first consider “nonunital colagebras” over \(\Phi \), and then treat unitality as a condition and not a structure. To do this, let P be the category with one object o and two morphisms, \(\mathsf{id}\) and p, subject to the relation \(p^2=p\). Then any stable comonad \(\Phi \) on a stable model pair \(\langle \mathcal {C},\mathcal {C}' \rangle \) defines a stable family \(\langle R(\mathcal {C},\Phi ),R(\mathcal {C}',\Phi ) \rangle \) over P, with fiber \(\langle \mathcal {C},\mathcal {C}' \rangle \) and transition functor \(p_! = P\). We then consider the corresponding category \({\text {DRec}}(P,R(\Phi ,P))\) whose objects correspond to derived nonunital coalgebras over \(\Phi \).
Alternatively, let \(\overline{\Delta } \subset \Delta \) be the subcategory with the same objects an injective maps between them. Then the reflection \(S(\mathcal {C}',\Phi )^\flat \) of the barycentric expansion \(S(\mathcal {C}',\Phi )\) gives by restriction a model prefibration \(\widetilde{\mathcal {C}}'\) over \(\overline{\Delta }^o\), and since \(\overline{\Delta }^o\) is an ordered category with no latching maps, this prefibration is tautologically good. We can then define the subcategory \(\widetilde{\mathcal {C}} \subset \widetilde{\mathcal {C}}'\) as in Definition 9.4, and \(\langle \widetilde{\mathcal {C}},\widetilde{\mathcal {C}}' \rangle \) is a stable model prefibration over \(\overline{\Delta }^o\) in the sense of Definition 8.1. Say that a map f in \(\overline{\Delta }^o\) is special if it is special in \(\Delta ^o\), say that a derived section \(E \in {\text {DSec}}(\overline{\Delta }^o,\widetilde{\mathcal {C}})\) is homotopy special if it is homotopy cartesian over all special maps, and let \({\text {DSec}}_+(\overline{\Delta }^o,\widetilde{\mathcal {C}}) \subset {\text {DSec}}_+(\overline{\Delta }^o,\widetilde{\mathcal {C}})\) be the full subcategory spanned by homotopy special maps.
Lemma 10.1
Proof
Consider for a moment the special situation \(\Phi = \mathsf{Id}\), the trivial comonad, and assume that \(\mathcal {C}'\) has countable limits. Then the stable model family \(R(\mathcal {C}',\Phi )\) is within the scope of Proposition 9.10, so that the category of Lemma 10.1 is equivalent to the homotopy category \({\text {Ho}}(\mathcal {C}^P)\). Objects in \(\mathcal {C}^{'P}\) are pairs \(\langle E,p \rangle \) of an object \(E \in \mathcal {C}'\) and an idempotent \(p:E \rightarrow E\), \(p^2 = p\). We have a natural embedding \(e:\mathcal {C}' \rightarrow \mathcal {C}^{'P}\) sending \(E \in \mathcal {C}'\) to \(\langle E,\mathsf{id}\rangle \), and since \(\mathcal {C}'\) has countable limits, it also has images of idempotents, so that e admits a left and rightadjoint functor \(e_\dagger :\mathcal {C}^{'P} \rightarrow \mathcal {C}'\) sending \(\langle E,p \rangle \) to the image of p. Both e and \(e_\dagger \) preserve weak equivalences, hence descend to an adjoint pair of functors of homotopy categories, and since e is fully faithful, \({\text {Ho}}(e)\) is also fully faithful. Restricting to \({\text {Ho}}(\mathcal {C}) \subset {\text {Ho}}(\mathcal {C}')\), we obtain a fully faithful embedding \(e:{\text {Ho}}(\mathcal {C}) \rightarrow {\text {Ho}}(\mathcal {C}^P)\).
Definition 10.2
A derived coalgebra over a stable comonad \(\Phi \) is a derived cosection \(E \in {\text {DRec}}(P,R(\mathcal {C},\Phi ))\) such that \(R^\cdot \varepsilon (E) \in {\text {Ho}}(\mathcal {C}^P)\) lies in the essential image of the fully faithful embedding \(e:{\text {Ho}}(\mathcal {C}) \rightarrow {\text {Ho}}(\mathcal {C}^P)\). The full subcategory in \({\text {DRec}}(P,R(\mathcal {C},\Phi ))\) spanned by derived coalgebras is denoted \({\text {DCoalg}}(\mathcal {C},\Phi ) \subset {\text {DRec}}(P,R(\mathcal {C},\Phi ))\).
Proposition 10.3
Assume given a stable pair \(\langle \mathcal {C},\mathcal {C}' \rangle \) such that the category \(\mathcal {C}'\) has countable limits, and assume that \(\mathcal {C}\subset \mathcal {C}'\) is closed under these limits. Then for any stable comonad \(\Phi \) on \(\langle \mathcal {C},\mathcal {C}' \rangle \), the functor \(e:{\text {DCoalg}}(\mathcal {C},\Phi ) \rightarrow {\text {Ho}}(\mathcal {C})\) admits a rightadjoint \(e^\dagger \), and the comonad \(e^\dagger \circ e\) on the category \({\text {Ho}}(\mathcal {C})\) is isomorphic to the comonad \(R^\cdot \Phi \).
Proof
10.2 A model for spectra
As an application of the technology of stable comonads, let us sketch a simple model for the stable homotopy category based on chain complexes.
Now denote by \({\text {Sets}}_+\) the category of pointed sets, and consider the forgetful functor \({\text {Ab}}\rightarrow {\text {Sets}}_+\) sending an abelian group E to its underlying set with distinguished element \(0 \in E\). This functor has a leftadjoint sending a pointed set \(\langle X,o \rangle \) to the quotient \(\mathsf{Span}(X) = {\mathbb {Z}}[X]/{\mathbb {Z}}\cdot o\) of the free abelian group \({\mathbb {Z}}[X]\) generated by X by the subgroup spanned by the distinguished element \(o \in X\). Thus \(E \rightarrow \mathsf{Span}(E)\) is a comonad on \({\text {Ab}}\), and this comonad obviously commutes with filtered colimits. Denote by \(Q = {\text {Stab}}(\mathsf{Span})\) the corresponding comonad on \(C_\cdot ({\text {Ab}})\), and equip \(C_\cdot ({\text {Ab}})\) with the projective model structure of Example 2.13, so that \(\langle C_\cdot ({\text {Ab}}),C_\cdot ({\text {Ab}}) \rangle \) is a stable model pair.
Lemma 10.4
The comonad Q on \(\langle C_\cdot ({\text {Ab}}),C_\cdot ({\text {Ab}}) \rangle \) is stable in the sense of Definition 10.2.
Proof
By definition, Q commutes with filtered colimits. On one hand, a filtered colimit of quasiisomorphisms resp. fibrations in \(C_\cdot ({\text {Ab}})\) is a quasiisomorphism resp. fibration, and on the other hand, any fibration resp. trivial fibration f in \(C_\cdot ({\text {Ab}})\) can be represented as a filtered colimit of fibrations resp. trivial fibrations \(f_n:E_\cdot \rightarrow E_\cdot '\) such that \(E_m=E_m'=0\) for \(m \gg 0\). Thus it suffices to prove that for any n, Q is a stable comonad on the full subcategory \(C_{\le n}({\text {Ab}}) \subset C_\cdot ({\text {Ab}})\) of complexes \(E_\cdot \) with \(E_m=0\) for \(m \ge n\). Since Q commutes with homological shifts, it actually suffices to consider the case \(n=0\), when \({\text {Stab}}\) is given by (10.5). Then the fact that Q sends fibrations to fibrations immediately follows from the fact that \(\mathsf{Span}\) sends surjective maps to surjective maps, and the fact that Q is stable and sends quasiisomorphisms to quasiisomorphisms is wellknown. \(\square \)
Lemma 10.5
 (i)
Assume given an object \(S \in {\text {DCoalg}}(C_\cdot ({\text {Ab}}),Q)\) such that \(e(S) = {\mathbb {Z}}\in {\text {Ab}}\subset {\mathcal {D}}({\text {Ab}})\). Then \({\text {Hom}}(S,E)=0\) for any \(E \in {\text {DCoalg}}^{\le 1}(C_\cdot ({\text {Ab}}),Q)\).
 (ii)
The subcategories \({\text {DCoalg}}^{\le n}(C_\cdot ({\text {Ab}}),Q)\), \(n \in {\mathbb {Z}}\) form a nondegenerate tstructure on \({\text {DCoalg}}^(C_\cdot ({\text {Ab}}),Q)\). The forgetful functor e and its adjoint functor \(e^\dagger \) are right texact. The functor e induces an equivalence between the heart of the tstructure on \({\text {DCoalg}}^(C_\cdot ({\text {Ab}}),Q)\) and the heart \({\text {Ab}}= {\mathcal {D}}^{\le 0}({\text {Ab}}) \cap {\mathcal {D}}^{\ge 0}({\text {Ab}}) \subset {\mathcal {D}}({\text {Ab}})\) of the standard tstructure on \({\mathcal {D}}({\text {Ab}})\).
Proof
For (i), note that for any complex \(E_\cdot \in C_\cdot ({\text {Ab}})\), the counit map \(Q(E_\cdot ) \rightarrow E_\cdot \) is surjective, and let \(\overline{Q}(E_\cdot )\) be its kernel. Then \(\overline{Q}\) is a nonunital comonad on \(C_\cdot ({\text {Ab}})\), it preserves quasiisomorphisms, and it is a wellknown property of the functor Q that \(\overline{Q}\) sends \({\mathcal {D}}^{\le n}({\text {Ab}})\) into \({\mathcal {D}}^{\le n1}({\text {Ab}})\) for any integer n. Then to compute \({\text {Hom}}(S,E)\), we can use the fundamental spectral sequence of Corollary 8.11. Its \(E_1\)term is the standard cobar complex of the unital comonad Q, and it is quasiisomorphic to the cobar complex of the nonunital comonad \(\overline{Q}\) with terms \({\text {Hom}}({\mathbb {Z}},\overline{Q}^n(e(E))[n])\). All these terms vanish for dimension reasons.
For (ii), note that the functor e is right texact by definition, and by construction, \(e^\dagger \circ e \cong {\text {Ho}}(Q)\) is also right texact; since e is conservative, this means that \(e^\dagger \) is right texact. By induction, to prove that the categories \({\text {DCoalg}}^{\le n}(C_\cdot ({\text {Ab}}),Q)\) form a nondegenerate tstructure, it suffices to prove that for any integer n, the embedding \({\text {DCoalg}}^{\le n1}(C_\cdot ({\text {Ab}}),Q) \subset {\text {DCoalg}}^{\le n}(C_\cdot ({\text {Ab}}),Q)\) admits a rightadjoint. This is stable under shifts, so it suffices to take \(n=0\). Then for any \(E \in {\text {DCoalg}}^{\le 0}(C_\cdot ({\text {Ab}}),Q)\), the object \(e^\dagger (\tau _{\ge 0}e(E))\) is rightorthogonal to \({\text {DCoalg}}^{\le 1}(C_\cdot (A\mathsf{B}),Q)\) by adjunction, so it suffices to prove that the cone of the adjunction map \(E \rightarrow e^\dagger (\tau _{\ge 0}e(E))\) lies in \({\text {DCoalg}}^{\le 1}(C_\cdot ({\text {Ab}}),Q)\). This, together with the claim about the hearts, amounts to checking that for any \(E \in {\mathcal {D}}^{\le 0}({\text {Ab}})\), the adjunction map \(Q(M) = e(e^\dagger (E)) \rightarrow E\) is an isomorphism in homological degree 0. This is equivalent to saying that \(\overline{Q}\) sends \({\mathcal {D}}^{\le 0}({\text {Ab}})\) into \({\mathcal {D}}^{\le 1}({\text {Ab}})\). \(\square \)
10.3 Comparison
Now let us show how that category \({\mathcal {D}}(\mathcal {C}_\cdot ({\text {Ab}}),Q)\) is related to spectra. Recall that a spectrum is what represents a generalized cohomology theory, and in the most classic model, it is given by a collection \(\{X_\cdot \}\) of pointed simplicial sets \(X_n\), \(n \ge 0\) equipped with maps \(\delta _n:\Sigma X_n \rightarrow X_{n+1}\), where \(\Sigma X = S^1 \wedge X\) is the suspension functor (smashproduct with the standard simplicial circle \(S^1\)). A map between two such collections \(\{X_\cdot \}\), \(\{Y_\cdot \}\) is a collection of maps \(X_n \rightarrow Y_n\) that commute with the maps \(\delta _n\), a map is a weak equivalence iff all the component maps \(X_n \rightarrow Y_n\) are weak equivalences, and inverting weak equivalences via any of the many model structures present in the literature, one obtains the triangulated stable homotopy category \(\mathsf{StHom}\).
The category \(\mathsf{StHom}\) has arbitrary sums, and it has the following remarkable property: for any triangulated category \({\mathcal {D}}\) with arbitrary sums, a triangulated functor \(\nu :\mathsf{StHom}\rightarrow {\mathcal {D}}\) that commutes with arbitrary sums admits a rightadjoint \(\nu ^\dagger \) (this is a part of Brown Representability Theorem).
Theorem 10.6
The functor (10.12) induces an equivalence of categories between \(\mathsf{StHom}^ \subset \mathsf{StHom}\) and \({\text {DCoalg}}^(C_\cdot ({\text {Ab}}),Q) \subset {\text {DCoalg}}(C_\cdot ({\text {Ab}}),Q)\).
Proof
By definition, we have \(H \cong e \circ h\), where \(e:{\text {DCoalg}}(C_\cdot ({\text {Ab}}),Q) \rightarrow {\mathcal {D}}({\text {Ab}})\) is the forgetful functor. The functor e has a rightadjoint functor \(e^\dagger \), thus commutes with arbitrary sums, and then since e is conservative, this implies that h also commutes with arbitrary sums. Therefore by Brown representability, it has a rightadjoint functor \(h^\dagger \), and we have \(H^\dagger \cong h^\dagger \circ e^\dagger \).
Remark 10.7
Since the category \({\text {DCoalg}}(C_\cdot ({\text {Ab}}),Q)\) comes from a stable model pair, it has a spectral enrichment of Lemma 9.8. It is not difficult to show that it is compatible with the equivalence (10.12); we do not do it to save space.
Remark 10.8
Theorem 10.6 is of course only useful if one can get good control over the functor Q, and it looks like it is much too big for that. Thus we mostly include Theorem 10.6 as a proofofconcept and an illustration for Definition 10.2. We expect derived coalgebras to be much more useful when comonads in questions are polynomial functors (such as the cyclic power functor considered in [28]).
11 DG categories
11.1 Linear structures
If a DG category \(A_\cdot \) is small, then modules over \(A_\cdot \) form a category; denote it by \(C_\cdot (A_\cdot )\). The category \(C_\cdot (A_\cdot )\) has a natural projective model structure whose weak equivalences W resp. fibrations F are pointwise quasiisomorphisms resp. pointwise degreewise surjections. Denote by \(C^{pf}_\cdot (A_\cdot ) \subset C_\cdot (A_\cdot )\) the smallest full subcategory closed under weak equivalences, shifts and cones, and containing all the modules \(\mathsf{Y}(a)\), \(a \in A_\cdot \). It is elementary to check that \(C_\cdot (A_\cdot )\) is a stable model category, and \(\langle C^{pf}_\cdot (A_\cdot ),C_\cdot (A_\cdot ) \rangle \) is a stable model pair in the sense of Definition 2.12. The triangulated category \({\text {Ho}}(C_\cdot (A_\cdot ),W)\) is denoted by \({\mathcal {D}}(A_\cdot )\) and called the derived category of \(A_\cdot \)modules, and \({\mathcal {D}}^{pf}(A_\cdot ) = {\text {Ho}}(C^{pf}_\cdot (A_\cdot )) \subset {\mathcal {D}}(A_\cdot )\) is a full triangulated subcategory whose Karoubi envelope is the subcategory of compact objects in \({\mathcal {D}}(A_\cdot )\). If \(A_\cdot \) is pretriangulated, then the Yoneda functor \(a \mapsto \mathsf{Y}(a)\) induces a triangulated equivalence \(H_0(A_\cdot ) \cong {\mathcal {D}}^{pf}(A_\cdot )\).
More generally, if a pretriangulated DG category \(A_\cdot \) is not small, it often happens that it contains a small full subcategory \(A'_\cdot \) such that the Yoneda functor \(H^0(A_\cdot ) \rightarrow {\mathcal {D}}(A'_\cdot )\), \(a \mapsto \mathsf{Y}(a)_{A'_\cdot }\) is fully faithful. In this case, \(H_0(A_\cdot )\) also corresponds to a stable model pair—the one formed by \(C_\cdot (A'_\cdot )\) and its full subcategory spanned by objects weakly equivalent to \(\mathsf{Y}(a)\), \(a \in A_\cdot \).
Conversely, assume given a stable model pair \(\langle \mathcal {C},\mathcal {C}' \rangle \). Fix a commutative ring k, and denote by \(C^{pf}_\cdot (k) \subset C_\cdot (k)\) the full subcategory spanned by finitelength complexes of finitely generated free kmodules. Note that \(C^{pf}_\cdot (k) \subset C_\cdot (k)\) is a monoidal subcategory closed under finite coproducts, with the unit object k placed in degree 0. Moreover, \(C_\cdot (k)\) carries the projective model structure, and all objects \(M_\cdot \in C^{pf}_\cdot (k) \subset C_\cdot (k)\) are fibrant and cofibrant with respect to this model structure.
Definition 11.1
 (i)
for any \(M_\cdot \), the functor \(M_\cdot \otimes :\mathcal {C}' \rightarrow \mathcal {C}'\) sends \(\mathcal {C}\subset \mathcal {C}'\) into \(\mathcal {C}\subset \mathcal {C}'\) and preserves finite limits and classes C, F, W, and
 (ii)
for any fibrant and cofibrant \(X \in \mathcal {C}\), the functor \( \otimes X:C^{pf}_\cdot (k) \rightarrow \mathcal {C}\subset \mathcal {C}'\) preserves finite products and classes C, F, W.
Example 11.2
For any small DG category \(A_\cdot \) over k, pointwise tensor product gives a canonical klinear structure on \(\langle C^{pf}_\cdot (A_\cdot ),C_\cdot (A_\cdot ) \rangle \).
 (i)
objects of \(\mathcal {C}_\cdot \) are fibrant and cofibrant objects \(X \in \mathcal {C}\subset \mathcal {C}'\),
 (ii)for any two objects \(X,Y \in \mathcal {C}_\cdot \), the complex \(\mathcal {C}_\cdot (c,c')\) is given bywhere I is as in (11.2), the commutative ring k acts via its action on I, and the differential \(d:\mathcal {C}_{i+1}(X,Y) \rightarrow \mathcal {C}_i(X,Y)\) is induced by the natural map \(\alpha [(i+1)] \circ \beta [i]:I[i] \rightarrow I[(i+1)]\).$$\begin{aligned} \mathcal {C}_i(X,Y) = \mathcal {C}(X,I[i] \otimes Y), \qquad i \in {\mathbb {Z}}, \end{aligned}$$
Proposition 11.3
The DG category \(\mathcal {C}_\cdot \) is pretriangulated, and the triangulated category \(H_0(\mathcal {C}_\cdot )\) is naturally equivalent to \({\text {Ho}}(\mathcal {C},W)\) as a triangulated category.
Proof
Finally, to see that the equivalence is triangulated, recall that cones in \(H_0(\mathcal {C}_\cdot )\) are defined by cartesian and homotopy cartesian squares (11.4), and these canonically extend to the fiber versions of diagrams (2.11). \(\square \)
11.2 Gluing
It turns out that the notion of a klinear structure is not strong enough for the gluing formalism of Sects. 3 and 9, and we have to modify it. Note that for any category \(\mathcal {C}\) and small category I, a functor \(C_\cdot ^{pf}(k) \times \mathcal {C}\rightarrow \mathcal {C}\) induces a functor \(C_\cdot ^{pf}(k) \times \mathcal {C}^I \times \mathcal {C}^I\) for the functor category \(\mathcal {C}^I\).
Definition 11.4
A klinear structure on a stable model pair \(\langle \mathcal {C},\mathcal {C}' \rangle \) is left resp. rightcontinuous if the induced functor \(C_\cdot ^{pf}(k) \times \mathcal {C}^{[1]} \rightarrow \mathcal {C}^{[1]}\) is a klinear structure on the stable model pair \(\langle \mathcal {C}^{[1]},\mathcal {C}^{'[1]} \rangle \), where \(\mathcal {C}^{'[1]}\) is equipped with the projective resp. injective model structure, and continuous if it is left and rightcontinuous.
Although it is not strictly necessary, it is also convenient to strengthen Definition 8.1.
Definition 11.5
A k linear stable model prefibration \(\langle \mathcal {C},\mathcal {C}' \rangle \) over a category I is a stable model prefibration \(\langle \mathcal {C},\mathcal {C}' \rangle \) over I in the sense of Definition 8.1 such that for any morphism \(f:i \rightarrow i'\) in I, the transition functor \(f^*:\mathcal {C}'_{i'} \rightarrow \mathcal {C}_i\) preserves finite limits and sends \(\mathcal {C}_{i'} \subset \mathcal {C}'_{i'}\) into \(\mathcal {C}_i \subset \mathcal {C}'_i\), equipped with a functor \(\otimes :C_\cdot ^{pf}(k) \times \mathcal {C}' \rightarrow \mathcal {C}'\) cartesian over I, and associativity and unitality isomorphisms such that for any \(i \in \mathcal {C}_i\), the induced functor \(\otimes :\mathcal {C}_\cdot ^{pf}(k) \times \mathcal {C}'_i \rightarrow \mathcal {C}'_i\) is a continuous klinear structure on \(\langle \mathcal {C}_i,\mathcal {C}'_i \rangle \) in the sense of Definition 11.4.
We note that since \(f^*(\mathcal {C}_{i'}) \subset \mathcal {C}_i\) for any \(f:i \rightarrow i'\), the functor \(\mathcal {C}\rightarrow I\) is also a prefibration, so that saying that \(\otimes \) is cartesian over I makes sense. One probably could get away with weaker assumptions on \(f^*\), but we do not know any interesting examples that would require larger generality. In particular, we do need the transition functors \(f^*\) to commute with \(\otimes \), and this already implies that they commute with finite sums and products. One immediate simplification that results from requiring the functors \(f^*\) to preserve finite limits is that if I is a Reedy category and \(\langle \mathcal {C},\mathcal {C}' \rangle \) is good in the sense of Definition 8.1, then \(\mathcal {C}'\) satisfies the condition (i) of Theorem 7.17. Therefore there is no need to consider matching expansion—by Propositions 8.2 and 7.26, already \(\langle {\text {Sec}}(I,\mathcal {C}),{\text {Sec}}(I,\mathcal {C}') \rangle \) is a stable model pair.
Proposition 11.6
Assume given a stable model prefibration \(\langle \mathcal {C},\mathcal {C}' \rangle \) over a Reedy category I that is good in the sense of Definition 8.1 and klinear in the sense of Definition 11.5. Then the functor \(\otimes \) induces a continuous klinear structure in the sense of Definition 11.4 on the stable model pair \(\langle {\text {Sec}}(I,\mathcal {C}),{\text {Sec}}(I,\mathcal {C}') \rangle \).
Proof
Definition 11.1 (i) immediately follows from the definitions. To prove that \(\otimes \) gives a klinear structure, one has to check is Definition 11.1 (ii)—namely, the fact that \(\otimes \) sends fibrations and cofibrations in \(C_\cdot ^{pf}(k)\) to fibrations and cofibrations in \({\text {Sec}}(I,\mathcal {C})\). The category \(C_\cdot ^{pf}(k)\) is selfdual, and the duality interchanges fibrations and confibrations, so that it suffices to prove the statement for fibration. Moreover, by induction on degree and Proposition 3.12, it suffices to do it in the situation of Proposition 3.8, under the assumption that the functor \(\Phi \) sends \(C_0 \subset \mathcal {C}_0'\) into \(\mathcal {C}_1 \subset \mathcal {C}_1'\) and preserves finite limits. Then an object \(\langle c_0,c_1,\alpha \rangle \in \mathsf{R}(\Phi )\) is fibrant cofibrant iff \(c_0\) is fibrant cofibrant in \(\mathcal {C}_0\) and \(\alpha :c_1 \rightarrow \Phi (c_0)\) is fibrant cofibrant in \(\mathcal {C}_1^{[1]}\), so that the claim immediately follows from Definition 11.1 (ii) for \(\mathcal {C}_0\) and \(\mathcal {C}_1^{[1]}\). Thus \(\otimes \) indeed gives a klinear structure; to prove that it is continuous, apply the same argument to \(I \times [1]\). \(\square \)
Once we have fixed Definition 11.5, giving a klinear version of stable model families of Definition 9.3 and stable comonads of Section 9.3 is completely straightforward. For a stable model family \(\langle \mathcal {C},\mathcal {C}' \rangle \), we require that transition functor \(f_!\) for a map \(f:i \rightarrow i'\) sends \(\mathcal {C}_i\) into \(\mathcal {C}_{i'}\) and preserves finite limits, and that the functor \(\otimes \) defining a klinear structure is cocartesian over I. For a comonad \(\Phi \) on a stable klinear model pair \(\langle \mathcal {C},\mathcal {C}'\rangle \), we require that \(\Phi (\mathcal {C}) \subset \mathcal {C}\), \(\Phi \) preserves finite limits, and we have an isomorphism \(\Phi \circ \otimes \cong \otimes \circ (\mathsf{Id}\times \Phi )\) that is compatible with all the structure maps. Under these assumptions, Proposition 11.6 produces klinear structures on the stable model pairs defining the categories \({\text {DRec}}(I,\mathcal {C})\) resp. \({\text {DCoalg}}(\mathcal {C},\Phi )\). All the general theory works in the klinear case, too; in particular, we have the spectral sequence of Corollary 8.11.
Remark 11.7
An example of a klinear comonad on the category \(C_\cdot (k)\) is given by \(\Phi (M_\cdot ) = {\text {Hom}}_k(A_\cdot , M_\cdot )\), where \(A_\cdot \) is a DG algebra termiseprojective over k, and it is easy to see that under mild finiteness assumptions, all examples are of this form. In this case, coalgebras over \(\Phi \) coincide with modules over \(A_\cdot \), the fundamental spectral sequence of Corollary 8.11 reduces to the bar complex of the algebra \(A_\cdot \), and derived coalgebras in our sense roughly correspond to \(A_\infty \)modules. One can probably also incorporate \(A_\infty \)algebras into the picture by considering weakly special klinear prefibration over \(\Delta \) but we did not go into this. Note that the fact that unitality requires special treatment and is best considered simply as a condition on the level of the derived category is wellknown in the theory of \(A_\infty \)algebras and \(A_\infty \)modules (see e.g. [36], where the combinatorics is very close to our Lemma 10.1).
11.3 Examples
Apart from DG categories, the main source of klinear stable model pairs are klinear abelian categories. If we have a klinear abelian category \(\mathcal {E}\) with enough projectives resp. injectives, then the category of chain complexes \(C_\cdot (\mathcal {E})\) with the projective resp. injective model structure of Example 2.13 is a stable model category. For any abelian subcategory \(\mathcal {E}_0 \subset \mathcal {E}\), we can consider the full subcategory \(C_\cdot (\mathcal {E},\mathcal {E}_0) \subset C_\cdot (\mathcal {E})\) spanned by complexes with homology in \(\mathcal {E}_0\), and then \(\langle C_\cdot (\mathcal {E},\mathcal {E}_0),C_\cdot (\mathcal {E}) \rangle \) is a klinear stable model pair. If we denote by \(C^{pf}_\cdot (\mathcal {E}) \subset C_\cdot (\mathcal {E})\) the full subcategory spanned by perfect complexes, then \(\langle C_\cdot ^{pf}(\mathcal {E}),C_\cdot (\mathcal {E}) \rangle \) is also a klinear stable model pair.
For example, for any Noetherian scheme X / k, the category \({\text {QCoh}}(X)\) of quasicoherent sheaves on X has enough injectives. If X is affine, the subcategory \({\text {Coh}}(X) \subset {\text {QCoh}}(X)\) has enough projectives. In general, it does not, but it is still a full abelian subcategory in \({\text {QCoh}}(X)\), so that we have a klinear stable model pair \(\langle C_\cdot ({\text {QCoh}}(X),{\text {Coh}}(X)),C_\cdot ({\text {QCoh}}(X)) \rangle \). The corresponding triangulated category is the derived category \({\mathcal {D}}_c(X)\) of complexes with coherent homology, and it is in fact equivalent to the derived category \({\mathcal {D}}({\text {Coh}}(X))\). Analogously, the klinear stable model pair \(\langle C^{pf}_\cdot ({\text {QCoh}}(X)),C_\cdot ({\text {QCoh}}(X))\rangle \) produces the derived category \({\mathcal {D}}^{pf}(X)\) of perfect complexes on X.
The corresponding pretriangulated DG categories of Proposition 11.3 are quasiequivalent to the standard DG enhancements for \({\mathcal {D}}_c(X)\) and \({\mathcal {D}}^{pf}(X)\), and as far as categories are concerned, the theories seem completely parallel. However, they diverge when we start dealing with functors. Namely, consider the following situation.
Example 11.8
Assume given two klinear abelian categories \(\mathcal {E}_0\), \(\mathcal {E}_1\) with enough projectives resp. injectives, and a pair of klinear functors \(e:\mathcal {E}_0 \rightarrow \mathcal {E}_1\), \(e^\dagger :\mathcal {E}_1 \rightarrow \mathcal {E}_0\) rightadjoint to e. Then by adjunction, e is rightexact and \(e^\dagger \) is leftexact, and if we equip \(C_\cdot (\mathcal {E}_0)\) resp. \(C_\cdot (\mathcal {E}_1)\) with the projective resp. injective model structure, then termwise extensions of e and \(e^\dagger \) are left resp. rightderivable and form a Quillen adjoint pair.
The situation of Example 11.8 occurs in geometry when one considers partial and categorical resolutions, such as e.g. [31]. Namely, assume given an affine scheme Y / k and a proper map \(\pi :X \rightarrow Y\). Then the pullback functor \(e=\pi ^*\) and the pushforward functor \(e^\dagger =\pi _*\) as as in Example 11.8. We now note that neither \(L^\cdot \pi ^*\) nor \(R^\cdot \pi _*\) has a natural enhancement to a DG functor, so that in the standard DG formalism, one can only consider these functors if one makes additional choices. In the stable model pair formalism, no choices are needed. What happens is, the DG category associated to a stable model pair in Proposition 11.3 consists of fibrant cofibrant objects, but while these span the homotopy category, they are not necessarily preserved by derived functors. Therefore not every stable left or rightderivable functor \(\Phi :\langle \mathcal {C},\mathcal {C}' \rangle \rightarrow \langle \mathcal {E},\mathcal {E}'\rangle \) between klinear stable model pairs descends to a DG functor \(\Phi _\cdot :\mathcal {C}_\cdot \rightarrow \mathcal {E}_\cdot \) between the corresponding DG categories. In this respect, the formalism of stable model pairs is more flexible. Another example of this flexibility is the FourierMukai transform of Example 3.5.
While functors of the form considered in Example 11.8 can be used in the elementary gluing construction of Proposition 3.8, their construction relies on using model structures of two different type on the source and the target, and this cannot be iterated. Here are two geometric examples of iterated gluing that use only one of the two model structures.
Example 11.9
Assume given an affine simplicial scheme X over k. Then for any morphism \(f:[n'] \rightarrow [n]\) in \(\Delta \), we have the corresponding affine map \(X(f):X([n]) \rightarrow X([n'])\), and the functor \(X(f)_*\) is exact, hence rightderivable with respect to the projective model structures. Then the categories \(C_\cdot ({\text {QCoh}}(X([n])))\) with the functors \(X(f)_*\) form a klinear stable model prefibration over \(\Delta \) in the sense of Definition 11.5.
Example 11.10
Assume given a small category I and a functor X from I to schemes. Then for any morphism \(f:i \rightarrow i'\) in I, we have the map \(X(f):X(i) \rightarrow X(i')\), and the functor \(X(f)_*\) is rightderivable with respect to the injective model structures. The categories \(C_\cdot ({\text {QCoh}}(X(i)))\) with the transition functors \(X(f)_*\) form a klinear stable model family over I.
In both Examples 11.9 and 11.10, one can of course restrict one’s attention to complexes of coherent sheaves, or perfect complexes, by considering the corresponding stable model pair (although one has to check that the transition functors preserve complexes in the chosen class). In Example 11.10, the category \({\text {DRec}}(I,C_\cdot ({\text {QCoh}}(X(i))))\) corresponds pretty closely to the intuitive idea of the derived category of quasicoherent sheaves on a diagram of schemes. In the interests of full disclosure, we should mention that it is probably equivalent to the derived category \({\mathcal {D}}({\text {Rec}}(I,{\text {QCoh}}(X(i)))\) of the abelian category \({\text {Rec}}(I,{\text {QCoh}}(X(i)))\), although we did not check this.
For another series of examples, one can consider topological sheaves instead of coherent ones. For the simplest example, let X be a simplicial scheme over \(\mathbb {C}\), and consider the categories \({\text {Shv}}(X([n])_{an},k)\) of sheaves of kmodules on the underlying complexanalytic spaces \(X([n])_{an}\). Then for any \(f:[n'] \rightarrow [n]\), the pullback functor \(X(f)^*:{\text {Shv}}(X([n'])_{an},k) \rightarrow {\text {Shv}}(X([n])_{an},k)\) is exact, thus rightderivable with respect to the injective model structure, and the categories \(C_\cdot ({\text {Shv}}(X([n])_{an},k))\) with the transition functors \(X(f)^*\) form a klinear stable model prefibration over \(\Delta ^o\). Again, one can restrict one’s attention to a subclass of sheaves by considering model pairs (for example, one can consider complexes of sheaves with constructible cohomology). The category \({\text {DSec}}(\Delta ^o,{\text {Shv}}(X_{an},k))\) then gives a good notion of the category of complexes of sheaves on the simplicial scheme X. One natural example of a simplicial scheme is the stack quotient \(Y/\!/G\) of a scheme Y by an action of an algebraic group G, understood as the simplicial scheme with terms \(X([n]) = Y \times G^n\). In this case, \({\text {DSec}}(\Delta ^o,{\text {Shv}}(X_{an},k))\) contains as a full subcategory the equivariant derived category \({\text {Shv}}_G(Y_{an},k)\) of [5].
12 Localization

\({\mathcal {D}}_0\) is the bounded derived category of finitedimensional vector spaces over a field k, and \({\mathcal {D}}\supset {\mathcal {D}}_0\) is the derived category of all vector spaces (or at least, of vector spaces of countable dimension).
12.1 Generalities
Historically, the first localization construction in the model category setting was given by Bousfield [7] who wanted to avoid enlarging universes when localizing the stable homotopy category. While [7] only deals with spaces and spectra, the approach has been generalized to a large class of stable and unstable model categories, and this generalization is now known as “Bousfield localization”. Roughly speaking, one starts with a model category \(\mathcal {C}\), enlarges the class W of weak equivalences just as in Theorem 3.13, keeps one of the two classes C or F, say C, defines F by the lifting property, and then uses a general machine to show that under certain assumptions, this gives a new model structure on \(\mathcal {C}\). The most difficult part seems to be Definition 2.6 (iv), that is, the existence of fibrant replacements. These are obtained by transfinite induction using the socalled “small object argument”.
In principle, this could be done in our setting of stable model pairs, and due to the way we have formulated Definition 2.12, this is even simpler than the general case—no transfinite induction is necessary, and a fibrant replacement can be obtained in one step.
However, this does not solve our problem, since it does not cover all the applications that occur in real life. For example, a procedure of Bousfield type can never apply to the Calkin category example.
The problem is exactly fibrant replacements. Since one has enlarged W, hence also \(C \cap W\), F has shrunk, and fibrant objects are no longer fibrant with respect to the new model structure. To correct for this, one replaces an object X with the filtered colimit over an appropriate category I of objects \(X'\) equipped with a map \(f:X \rightarrow X'\) in the class \(C \cap W\). In the stable case, being in W means that the cone of f is in the subcategory \({\mathcal {D}}_0\), and the I must be large enough to be cofinal after projecting to the category W(X) of (3.8). In the Calkin case, this means that I is infinite, and the colimit is not in W(X) anymore.
A moment’s reflection shows that the problem is not specific to the Calkin category. Indeed, by its very definition, the Bousfield localization functor is the derived functor of \(\mathsf{Id}:\mathcal {C}\rightarrow \mathcal {C}\), and then by Quillen Adjunction Theorem, it comes equipped with an adjoint. In the stable situation, we are then back in the case of a semiorthogonal decomposition (and then if we work with stable pairs and not just stable model categories, the localization procedure is not needed at all since there is a simpler alternative).
 (I)
Definition 2.6 is too weak. One should simply impose whatever one wants for Bousfield localization, e.g. the existence of all limits, the fact that the category is “cofibrantly generated”, and so on.
 (II)
Definition 2.6 is too strong. One could envision a notion of a premodel structure on a saturated relative category \(\langle \mathcal {C},W \rangle \) consisting of two classes of maps \(\overline{C}\), \(\overline{F}\) such that both are closed under compositions, \(W = \overline{F} \cdot \overline{C}\), maps in \(\overline{F}\) admit pullbacks and are closed under pullbacks, and maps in \(\overline{C}\) admit pushouts and are closed under pushouts. This already allows one to show that maps in \({\text {Ho}}(\mathcal {C},W)\) are reduced to diagrams (2.1) of length 3, and gives some control over the localization (in particular, one can impose a version of the stability condition and then prove that \({\text {Ho}}(\mathcal {C},W)\) is triangulated).
However, completely dispensing with model categories and developing a replacement such as (II) above from scratch would be a huge enterprise, and at present, we are not ready to do it.
So, in this paper, we adopt a third option. We keep the foundations exactly as they are, but we recall that we work with model pairs, and we use the freedom to enlarge the ambient category \(\mathcal {C}' \supset \mathcal {C}\). To do this, we need the technology of inductive completions.
12.2 Inductive completions
For any \(\langle I,X \rangle \in \mathsf{Ind}\,\mathcal {C}\) and cofinal functor \(\varphi :I' \rightarrow I\) with filtered \(I'\), the tautological map \(\langle I',\varphi ^*X \rangle \rightarrow \langle I,X \rangle \) is an isomorphism. In particular, for every filtered small category \(I'\), we have a cofinal functor \(I \rightarrow I'\) from a category I such that \(I^o\) is ordered in the sense of Definition 7.3 (for example, one can take the category \(I' = V(S)\) of Example 7.2, where S is the set of objects in I). Therefore every object in \(\mathsf{Ind}\,\mathcal {C}\) can be represented by a pair \(\langle I,X \rangle \) with ordered \(I^o\).
Lemma 12.1
Proof
Since \(\mathsf{Ind}\,\mathcal {C}\) has all filtered colimits, it suffices to check that it has colimits \(\mathsf{colim}_I\) over a finite ordered category I. These exists in \(\mathcal {C}\), so that we have an adjoint pair of functors \(\tau :\mathcal {C}\rightarrow \mathcal {C}^I\), \(\mathsf{colim}_I:\mathcal {C}^I \rightarrow \mathcal {C}\), with \(\tau \) sending \(c \in \mathcal {C}\) to the constant functor with value c. To extend this adjoint pair to \(\mathsf{Ind}\,\mathcal {C}\) and \((\mathsf{Ind}\,\mathcal {C})^I\), it suffices to use the equivalence (12.1). For finite limits, the argument is exactly the same. Finally, to check that (12.2) is an isomorphism, note that f can be represented as colimit over \(I' \times I\) of maps \(X_{i'} \times _{\langle I,X \rangle } X_i \rightarrow X_i\), and we first take colimit over \(I'\), we obtain exactly the map \(X_i \times _{\langle I,X \rangle } \langle I',X' \rangle \rightarrow X_i\). \(\square \)
Now assume given a saturated relative category \(\langle \mathcal {C},W \rangle \) with a model structure \(\langle C,F \rangle \). Note that since [1] is a finite ordered category, every map in \(\mathsf{Ind}\,\mathcal {C}\) can be represented by a filtered system \(\langle I,g \rangle \) of maps in \(\mathcal {C}\), and we can further assume that \(I^o\) is ordered.
Definition 12.2
A map in \(\mathsf{Ind}\,\mathcal {C}\) lies in the class C, F, \(\overline{C}\), \(\overline{F}\), \(\overline{W}\) iff it is a retract of a map given by a system \(\langle I,g \rangle \) with ordered \(I^o\) and g in the class C, F, \(C \cap W^I\), \(F \cap W^I\), \(W^I\) with respect to the model structure of Lemma 7.3 on \(\langle \mathcal {C}^I,W^I \rangle \). A map g in \(\mathsf{Ind}\,\mathcal {C}\) lies in the class W if \(g=f \circ c\) with \(f \in \overline{F}\) and \(c \in \overline{C}\).
Lemma 12.3
Assume given a commutative square (2.4) in \(\mathsf{Ind}\,\mathcal {C}\) such that \(f \in C\), \(f' \in \overline{F}\) or \(f \in \overline{C}\), \(f' \in F\). Then there exists a map \(q:Y \rightarrow X'\) such that \(g = q \circ f\) and \(g' = f' \circ q\).
Proof
Represent f as \(f = \mathsf{colim}_I\widetilde{f}\) with small filtered I with ordered \(I^o\), and \(\widetilde{f}:\widetilde{X} \rightarrow \widetilde{Y}\) in C resp. \(C \cap W^I\) with respect to the model structure of Lemma 7.3. Then by adjunction, it suffices to construct a splitting map \(q:\widetilde{Y} \rightarrow \tau ^*X'\) in \((\mathsf{Ind}\,\mathcal {C})^I\), where \(\tau :I \rightarrow \mathsf{pt}\) is the tautological projection. By induction on degree, it suffices to show that for any object \(i \in I\) of degree \(n = \deg (i)\), a splitting map q over \(I_{\le n1}\) can be extended to i. To do this, we can replace I with the full subcategory of objects \(i' \in I\) that admit a map \(i' \rightarrow i\), and by Definition 7.1 (ii)(c), this category is finite, hence also ordered (with the opposite degree function). Then by virtue of the isomorphism (12.1), we can represent \(\tau ^*(f')\) by a filtered colimit \(\mathsf{colim}_{I'}\widetilde{f}'\) of maps in \(\mathcal {C}^I\), and since \(\widetilde{f}\) is compact in \(\mathsf{Ind}\,\mathcal {C}^{I \times [1]}\), the map \(\widetilde{f} \rightarrow \tau ^*(f')\) given by g, \(g'\) factors through \(\widetilde{f}'(i)\) for some \(i' \in I'\). To extend the map q, it now suffices to apply the standard lifting property of the Reedy model structure on \(\mathcal {C}^I\). \(\square \)
Definition 12.4
A model category \(\langle \mathcal {C},W,C,F \rangle \) is rightproper if for any cartesian square (2.4) in \(\mathcal {C}\) with \(f' \in F\), \(g' \in W\), we have \(g \in E\).
Proposition 12.5
For any rightproper model category \(\langle \mathcal {C},W,C,F \rangle \), the inductive completion \(\mathsf{Ind}\,\mathcal {C}\) with the classes C, F, W of Definition 12.2 is a rightproper model category.
Proof
Definition 2.6 (i) is Lemma 12.1. By definition, we have \(\overline{C} \subset C \cap \overline{W}\) and \(\overline{F} \subset F \cap \overline{W}\). Any map in \(\mathsf{Ind}\,\mathcal {C}\) can be represented by a diagram \(\langle I,g \rangle \) such that \(I^o\) is ordered, and then Definition 2.6 (iv) for the category \(\mathcal {C}^I\) implies that it decomposes as \(f \circ c\) with \(f \in \overline{F}\), \(c \in C\) or with \(f \in F\), \(c \in \overline{C}\). If g is in \(W^I\), then in either of the decomposition, \(f \in \overline{F}\) and \(c \in \overline{C}\). Since \(\mathsf{Ind}\,\mathcal {C}\) has pullbacks, a retract of a composition \(f \circ c\) is the composition of retracts of f and c, so that this implies \(\overline{W} \subset W\) and \(\overline{F} \subset F \cap W\), \(\overline{C} \subset C \cap W\). Thus for a general \(\langle I,g \rangle \), we obtain Definition 2.6 (iv). Conversely, for any \(g = f \circ c \in W\) that also lies in C, the lifting property of Lemma 12.3 implies that g is a retract of c, thus lies in \(\overline{C}\), so that \(\overline{C} = C \cap W\). By the dual argument, \(\overline{F} = F \cap W\), and Definition 2.6 (iii) follows from Lemma 12.3. Moreover, by the standard argument, Definition 2.6 (iv) implies that a map g that has a left lifting property with respect to F resp. \(\overline{F}\) lies in \(\overline{C}\) resp. C, so that C and \(\overline{C}\) are closed under compositions and pushouts, and dually, F and \(\overline{F}\) and closed under compositions and pullbacks. This gives Definition 2.6 (ii).
To prove the Proposition, we now need to show that \(\langle \mathsf{Ind}\,\mathcal {C},W \rangle \) is a saturated relative category. So, assume given a composable pair of maps \(g_1:Y \rightarrow Z\), \(g_2:X \rightarrow Y\) in \(\mathsf{Ind}\,\mathcal {C}\), with composition \(g_{12} = g_1 \circ g_2\), and let us check the twooutofthree property of Definition 2.2.
If \(g_1,g_2 \in W\), then we need to prove that \(g_{12} \in W\), and since we already know that \(\overline{C}\) and \(\overline{F}\) are closed under compositions, it suffices to consider the case \(g_1 \in \overline{C}\), \(g_2 \in \overline{F}\). In this case, \(g_1\) can be represented by a filtered diagram \(\langle I,\widetilde{g}_1 \rangle \) with ordered \(I^o\) and \(\widetilde{g}_1:\widetilde{Y} \rightarrow \widetilde{Z}\) in \(C \cap W^I\), and since \(\overline{F}\) is stable under pullbacks, the induced map \(X \times _Y \widetilde{Y}(i) \rightarrow \widetilde{y}(i)\) is in \(\overline{F}\) for any \(i \in I\). Since W is obviously stable under filtered colimits, it suffices to prove that the composition map \(X \times _Y \widetilde{Y}(i) \rightarrow \widetilde{Z}(i)\) is in W for any i and apply the isomorphism (12.2) of Lemma 12.1. In other words, we may assume right away that \(g_2\) lies in \(\mathcal {C}\subset \mathsf{Ind}\,\mathcal {C}\). Now represent \(g_1\) as a filtered colimit of a diagram \(\langle I,\widetilde{g}_1 \rangle \) with another ordered small category \(I^o\) and \(\widetilde{g}_1:\widetilde{X} \rightarrow \widetilde{Y}\) pointwise in \(F \cap W^I\), and note that since \(Y \in \mathcal {C}\subset \mathsf{Ind}\,\mathcal {C}\) is compact in \(\mathsf{Ind}\,\mathcal {C}\), the isomorphism \(Y \cong \mathsf{colim}_I\widetilde{Y}\) factors through \(\widetilde{Y}(i)\) for some \(i \in I\). Then shrinking I if necessary, we may assume that the constant functor \(Y:I \rightarrow \mathcal {C}\) is a retract of \(\widetilde{Y}\), and replacing \(\widetilde{X}\) with \(\widetilde{X} \times _{\widetilde{Y}} Y\), we may further assume that \(\widetilde{Y}=Y\) is constant. Then \(g_1\) is represented by the constant map \(\widetilde{g}_1:Y \rightarrow Z\) between constant functors, and \(g_{12} = \mathsf{colim}_I \widetilde{g}_1 \circ \widetilde{g}_2\) is in \(\overline{W} \subset W\).
In the general case, decompose \(g_2 = f_2 \circ c_2\), \(g_{12} = f_{12} \circ c_{12}\) with \(c_2,c_{12} \in \overline{C}\), \(f_2,f_{12} \in \overline{F}\), and note that Lemma 12.3 provides a map q such that \(c_{12} = q \circ c_2\) and \(f_{12} \circ q = g_1 \circ f_2\). Then as we have just proved, we have \(q \in W\), so that \(g_1 \circ f_2 \in W\). Moreover, \(g_1 \in F\) by assumption, so that \(g_1 \circ f_2 \in F \cap W = \overline{F}\). Then as before, the isomorphism (12.2) reduces us to the case \(g_1 \in \mathcal {C}\subset \mathsf{Ind}\,\mathcal {C}\), and in this case, the claim follows from the fact that \(\langle \mathcal {C}^I,W^I \rangle \) is saturated by any small I.
Finally, assume that \(g_1,g_{12} \in W\), and let us prove that \(g_2 \in W\). Assume first that \(g_1,g_{12} \in \overline{F}\). Then we can decompose \(g_2 = f \circ c\), \(c \in \overline{C}\), \(f \in F\), and as we have already proved, we have \(g_1 \circ f \in W\), thus also \(g_1 \circ f \in \overline{F}\). Then by the same argument as before, proving that \(f \in W\) reduces to the corresponding property of the categories \(\mathcal {C}^I\), and therefore \(g_1 = f \circ c\) also lies in W.
Remark 12.6
It might very well be that Proposition 12.5 holds without assuming that \(\mathcal {C}'\) is rightproper, but unfortunately, we could not find a proof of this. As a mitigating circumstance, we note that model categories that appear in stable model pairs tend to be rightproper. For example, it is true for the category \(C_\cdot (\mathcal {C})\) of Example 2.13, both with the projective and injective model structure. Also, the model structure of Proposition 3.8 is obviously rightproper if the model structures on \(\mathcal {C}_0\), \(\mathcal {C}_1\) are rightproper and \(\Phi \) preserves weak equivalences, and then by induction, all the model structures of Theorem 7.17 are also rightproper, as soon as so are the fibers \(\mathcal {C}_i\), \(i \in I\) of the prefibration \(\mathcal {C}\), and the transition functors \(\mathcal {C}_i \rightarrow \mathcal {C}_{i'}\) preserve weak equivalences. This holds at least for the constant prefibrations of Sect. 9.3 with rightproper \(\mathcal {C}'\), and also for the stable comonad Q of Theorem 10.6.
12.3 Construction
Now assume given a stable model pair \(\langle \mathcal {C},\mathcal {C}' \rangle \), and assume that the model category \(\mathcal {C}'\) is rightproper in the sense of Definition 12.4.
Lemma 12.7
The pair \(\langle \mathsf{Ind}\,\mathcal {C},\mathsf{Ind}\,\mathcal {C}' \rangle \) is a stable model pair. The embedding \({\text {Ho}}(\mathcal {C}) \subset {\text {Ho}}(\mathsf{Ind}\,\mathcal {C})\) is fully faithful, the triangulated category \({\text {Ho}}(\mathsf{Ind}\,\mathcal {C})\) has arbitrary sums, any object \(X \in {\text {Ho}}(\mathcal {C})\) is compact in \({\text {Ho}}(\mathsf{Ind}\,\mathcal {C})\) (that is, \({\text {Hom}}(X,)\) commutes with arbitrary sums), and any full triangulated subcategory \({\mathcal {D}}\subset {\text {Ho}}(\mathsf{Ind}\,\mathcal {C})\) that contains \({\text {Ho}}(\mathcal {C})\) and is closed under all sums coincides with the whole \({\text {Ho}}(\mathsf{Ind}\,\mathcal {C})\).
Proof
The fact that \(\mathsf{Ind}\,\mathcal {C}\subset \mathsf{Ind}\,\mathcal {C}'\) is a model embedding immediately follows from Definition 12.2. Moreover, every cofiber resp. fiber square in \(\mathsf{Ind}\,\mathcal {C}'\) can be represented as \(\mathsf{colim}_I\) of a corresponding square in \(\mathcal {C}^{'I}\), with I filtered and \(I^o\) ordered, and conversely, \(\mathsf{colim}_I\) in such a situation sends weak equivalences to weak equivalences, fibration to fibration and cofibrations to cofibrations, and by Lemma 12.1, it also sends pushout resp. pullback squares to pushout resp. pullback squares. Then Definition 2.12 (i),(ii) immediately follow from the corresponding properties of \(\langle \mathcal {C}^I \subset \mathcal {C}^{'I} \rangle \) for a filtered I with ordered \(I^o\). Since \(\mathcal {C}\) has finite coproducts, \(\mathsf{Ind}\,\mathcal {C}\) has arbitrary coproducts, so that \({\text {Ho}}(\mathsf{Ind}\,\mathcal {C})\) has arbitrary sums. Such a sum can be represented by a filtered colimit of finite sums, so to check that \({\text {Hom}}(X,)\) commutes with sums, it suffices to represent X by a cofibrant object in \(\mathcal {C}\subset \mathcal {C}'\). Finally, any object \(X \in \mathsf{Ind}\,\mathcal {C}\) is a colimit \(\mathsf{colim}_I \widetilde{X}\), \(I^o\) ordered, and since \(\mathsf{colim}_I\) is right and leftderivable, we have \(h(X) \cong \mathsf{hocolim}_I h(\widetilde{X})\). This \(\mathsf{hocolim}_I\) can be computed for example by Proposition 9.13, and then it is clear that it lies in any triangulated subcategory in \({\text {Ho}}(\mathsf{Ind}\,\mathcal {C})\) that contains \({\text {Ho}}(\mathsf{Ind}\,\mathcal {C})\) and is closed under all sums. \(\square \)
Informally, one can say that \({\text {Ho}}(\mathsf{Ind}\,\mathcal {C})\) is the “triangulated inductive completion” of \({\text {Ho}}(C)\). However, this cannot be made formal: there is no general inductive completion procedure for triangulated categories, and in particular, \(\mathsf{Ind}\,{\mathcal {D}}\) is not triangulated for a general triangulated category \({\mathcal {D}}\). Our \({\text {Ho}}(\mathsf{Ind}\,\mathcal {C})\) depends on \(\langle \mathcal {C},W \rangle \), although it does not depend on the ambient model category \(\mathcal {C}' \supset \mathcal {C}\). Indeed, the class \(\overline{W}\) of maps in \(\mathsf{Ind}\,\mathcal {C}\) that appears in Definition 12.2 manifestly does not depend on \(\mathcal {C}'\), and it immediately follows from Definition 12.2 that \(\langle \mathsf{Ind}\,\mathcal {C},W \rangle \) is the saturation of the relative category \(\langle \mathsf{Ind}\,\mathcal {C},\overline{W} \rangle \).
Proposition 12.8
Assume given a stable model pair \(\langle \mathcal {C},\mathcal {C}' \rangle \) such that \(\mathcal {C}'\) is rightproper in the sense of Definition 12.4, and a full triangulated subcategory \({\mathcal {D}}_0 \subset {\mathcal {D}}= {\text {Ho}}(\mathcal {C})\) that is rightlocalizing in the sense of Definition 3.14. Let \(\langle \mathcal {C}_0,\mathcal {C}' \rangle \) be the corresponding stable model pair of Lemma 2.22. Then the embedding \(e:{\text {Ho}}(\mathsf{Ind}\,\mathcal {C}_0) \rightarrow {\text {Ho}}(\mathsf{Ind}\,\mathcal {C})\) is fully faithful and admits a rightadjoint functor \(e^\dagger \), and the full subcategory in \({\text {Ho}}(\mathsf{Ind}\,\mathcal {C})\) spanned by cones of the adjunction maps \(e(e^\dagger (E)) \rightarrow E\), \(E \in {\mathcal {D}}\subset {\text {Ho}}(\mathsf{Ind}\,\mathcal {C})\) is naturally equivalent to the quotient category \({\mathcal {D}}/{\mathcal {D}}_0\).
Proof
Notes
Acknowledgements
This paper was originally written for the anniversary volume celebrating the 70th birthday of my Ph.D. advisor David Kazhdan, and although it eventually grew much too long for that volume, it is still a pleasure to thank him for constant encouragement and attention to my work over all these years. Among a multitude of other things, it was he who first drew my attention to the question addressed in the paper, and it was he who always insisted I finish whatever I started. It is also a great pleasure to thank my Ph.D. student Eduard Balzin; he taught me a lot about the current state of the art in the subject, and his work forms a crucial part of the story developed here. A large part of this work was done while I was visiting CINVESTAV in Mexico City; it is a pleasure to acknowledge its hospitality and wonderful working atmosphere. Finally, it is a pleasure to thank the anonymous referee for a careful reading of the paper and many very useful suggestions.
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