Principal \(\infty \)bundles: presentations
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Abstract
We discuss two aspects of the presentation of the theory of principal \(\infty \)bundles in an \(\infty \)topos, introduced in Nikolaus et al. (Principal \(\infty \)bundles: general theory, 2012), in terms of categories of simplicial (pre)sheaves. First we show that over a cohesive site \(C\) and for \(G\) a presheaf of simplicial groups which is \(C\)acyclic, \(G\)principal \(\infty \)bundles over any object in the \(\infty \)topos over \(C\) are classified by hyperČechcohomology with coefficients in \(G\). Then we show that over a site \(C\) with enough points, principal \(\infty \)bundles in the \(\infty \)topos are presented by ordinary simplicial bundles in the sheaf topos that satisfy principality by stalkwise weak equivalences. Finally we discuss explicit details of these presentations for the discrete site (in discrete \(\infty \)groupoids) and the smooth site (in smooth \(\infty \)groupoids, generalizing Lie groupoids and differentiable stacks). In the companion article (Nikolaus et al. in Principal \(\infty \)bundles: examples and applications, 2012) we use these presentations for constructing classes of examples of (twisted) principal \(\infty \)bundles and for the discussion of various applications.
Keywords
Homotopy theory Topos theory Differential geometry Algebraic topology Principal bundles1 Overview
In [44] we have described a general theory of geometric principal \(\infty \)bundles (possibly twisted by local coefficients) and their classification by (twisted) nonabelian cohomology in \(\infty \)toposes. A certain charm of this theory is that, formulated the way it is in the abstract language of \(\infty \)topos theory, it is not only more general but also more elegant than the traditional theory. For instance every \(\infty \)group action is principal over its homotopy quotient, the quotient map is automatically locally trivial, the principal \(\infty \)bundle corresponding to a classifying map is simply its homotopy fiber (hence the universal principal \(\infty \)bundle is the point), and the fact that all principal \(\infty \)bundles arise this way is a fairly direct consequence of the axioms that characterize \(\infty \)toposes in the first place: the Giraud–Rezk–Lurie axioms.
While this abstract formulation provides a useful means to reason about general properties of principal \(\infty \)bundles, it is desirable to complement this with explicit presentations of the structures involved (notably of \(\infty \)groups, of \(\infty \)actions and of principal \(\infty \)bundles) by generators and relations. This is typically the way that explicit examples are constructed and in terms of which properties of these specific examples are computed in applications.
In recent years it has been well understood that the method of choice for presenting \(\infty \)categories by generators and relations is the homotopical category theory of categories of simplicial presheaves, i.e. presheaves of simplicial sets. The techniques themselves have a long history, dating back to work of Illusie [25], continued in the foundational work of [7] and developed further in [27, 32], which will play a prominent role below. Their interpretation as a generators and relations presentation for homotopy theoretic structures has been amplified in the exposition of [13], and was formalized in terms of model category theory by the main theorem in [14]. Finally [38] has provided the general abstract essence of this theorem in terms of the notion of presentable \(\infty \)categories. This is the notion of presentation that we are concerned with here.
 1.
Over a site \(C\) with a terminal object, every \(\infty \)group is presented by a presheaf of simplicial groups \(G\). (Proposition 3.35)
 2.
If the ambient \(\infty \)topos is locally \(\infty \)connected and local over an \(\infty \)cohesive site \(C\), and if \(G\) is \(C\)acyclic (Definition 3.43) then \(G\)principal \(\infty \)bundles over any object \(X\) are classified by simplicial hyperČechcohomology of \(X\) with coefficients in \(G\). In fact, the \(\infty \)groupoid of geometric \(G\)principal \(\infty \)bundles, morphisms and higher homotopies between these is equivalent to the \(\infty \)groupoid of Čech cocycles, Čech coboundaries and higher order coboundaries (Theorem 3.46).
 3.
If \(C\) is a site with enough points, then principal \(\infty \)bundles over \(C\) are presented by ordinary simplicial bundles in sheaves over \(C\) which satisfy a weakened notion of principality (Theorem 3.95).

the trivial site, modelling discrete geometry;

the site of smooth manifolds, modelling smooth/differential geometry.
Following the foundational work of Giraud [22], it seems that the first paper to consider the problem of giving a geometric description of nonabelian cohomology was the paper [17] of Duskin (this paper was intended as a precursor to a more substantial discussion, which unfortunately never materialized). This was followed by the more comprehensive treatment of Breen in [6]. This paper of Breen’s is noteworthy in that it treats nonabelian cohomology within the natural context of the homotopy theory of simplicial sheaves and it also introduces the notion of a pseudotorseur for a group stack; a notion which is closely related to our notion of weakly principal simplicial bundle. In [54, 55] Ulbrich gave a different interpretation of Duskin’s work, in particular introducing the notion of cocycle bitorsor which is closely related to Murray’s later notion of bundle gerbe [43]. Joyal and Tierney in [33] introduced a notion of pseudotorsor which is again closely related to our notion of weakly principal simplicial bundle; their notion of pseudotorsor was more general than the corresponding notion of Breen’s, since Breen restricted his attention to the case of 1truncated group objects while Joyal and Tierney worked with simplicial groupoids.
The mid 1990s saw a flurry of interest in interpreting geometrically the standard characteristic classes as higher principal bundles with structure \(\infty \)group of the form \(K(\pi ,n)\) for some abelian group \(\pi \); the works [8, 9, 10, 43] of Brylinski, Brylinski and McLaughlin and Murray are landmarks from this period. The overarching theme of these papers is to develop and apply a Chern–Weil theory for ‘higher line bundles’, in particular they focus attention on the \(\infty \)groups \({\mathbf {B}}U(1)\) and \({\mathbf {B}}^2U(1)\). Our aim here is less restrictive; we want to develop the theory of principal \(\infty \)bundles as a whole.
To continue the historical discussion, the 2004 thesis [3] gave a treatment of 1truncated principal \(\infty \)bundles—principal 2bundles—while [34] gave such a treatment from the point of view of bundle gerbes (we note that [34] appeared in preprint form in 2005). In [2] these constructions were generalized from structure 2groups to structure 2groupoids. The gauge 2groups of principal 2bundles were studied in [57]. A comprehensive account is given in [46].
Continuing in this vein, a discussion of 2truncated principal \(\infty \)bundles, principal 3bundles, was given in [35] in the guise of bundle 2gerbes, generalizing the abelian bundle 2gerbes (\({\mathbf {B}}^2 U(1)\)principal 3bundles) of [50].
The work that is closest to our discussion in Sect. 3.7.2 is the paper [31] of Jardine and Luo. This paper goes beyond the previous work of Breen [6] and Joyal and Tierney [33]; it introduces a notion of \(G\)torsor for \(G\) a group in \(\hbox {sPSh}(C)\) for some site \(C\) and shows that isomorphism classes of \(G\)torsors in \(\hbox {sPSh}(C)\) over \(*\) are in a bijective correspondence with the set of connected components of \( Maps (*,\overline{W}G)\). The presentation that we discuss in Sect. 3.7.2 is similar, differing in that it allows the base space to be an arbitrary simplicial presheaf and in that it reproduces the full homotopy type of the space of cocycles, not just their connected components.
Closely related also is the discussion in [49, 52], which is concerned with principal \(\infty \)bundles over topological spaces and in particular discusses their classification by traditional classifying spaces.
In summary, our work goes beyond that of all the works cited above in two directions; firstly we show that our notion of weakly principal bundle suffices to interpret the full homotopy type of the cocycle \(\infty \)groupoid, and secondly, we work over arbitrary bases: our base need not be just a space, it could be a 1stack or even an \(\infty \)stack, or differentiable versions of all of these (we remark that the study of bundles on differentiable stacks plays an important role in recent work on twisted \(K\)theory [20, 37]).
2 Presentations of \(\infty \)toposes
The presentations of principal \(\infty \)bundles and related structures in an \(\infty \)topos, discussed below in Sect. 3, builds on the presentation of the \(\infty \)topos itself by categories of simplicial (pre)sheaves. We assume the reader to be familiar with the basics of this theory (a good starting point is the appendix of [38], a classical reference is [18]), but in order to set up our notation and in order to record some statements, needed below, which are not easily found in the literature in the explicit form in which we will need them, we briefly recall some basics in Sect. 2.1. In Sect. 2.2 we discuss a general result about the representability of general objects in an \(\infty \)topos by simplicial objects in the site.
2.1 By simplicial presheaves
The monoidal functor \(\pi _0:\hbox {sSet}\rightarrow \hbox {Set}\) that sends a simplicial set to its set of connected components induces a functor \({\mathrm {Ho}}:{\mathrm {sSet}}{\mathrm {Cat}}\rightarrow {\mathrm {Cat}}\), where \({\mathrm {sSet}}{\mathrm {Cat}}\) denotes the category of \({\mathrm {sSet}}\)enriched categories [36]. Thus if \({\mathcal C}\) is an \({\mathrm {sSet}}\)enriched category then \({\mathrm {Ho}}({\mathcal C})\) is the category with the same underlying objects as \({\mathcal C}\) and with \({\mathrm {Ho}}({\mathcal C})(X,Y) := \pi _0 {\mathcal C}(X,Y)\) for all objects \(X, Y \in {\mathcal C}\). An \({\mathrm {sSet}}\)enriched functor \(f : {\mathcal C} \rightarrow {\mathcal D}\) is called a DKequivalence if \({\mathrm {Ho}}(f)\) is essentially surjective and if for all \(X,Y \in {\mathcal C}\) the morphism \(f_{X,Y} : {\mathcal C}(X,Y) \rightarrow {\mathcal D}(f(X), f(Y))\) is a weak homotopy equivalence. Write \(W_{\mathrm {DK}} \subset {\mathrm {sSet}}{\mathrm {Cat}}\) for the inclusion of the full subcategory whose morphisms are DKequivalences. This is a wide subcategory: an inclusion of categories that is bijective on objects.
For \(D\) a category and \(W \subset D\) a wide subcategory, to be called the subcategory of weak equivalences, the simplicial localization \(L_W D\) is the universal \({\mathrm {sSet}}\)enriched category with the property that morphisms in \(W \subset D\) become homotopy equivalences in \(L_W D\) [18]. For \(X, Y \in C\) two objects, the Kan complex \(L_W D(X,Y)\) is called the derived homspace or derived function complex or hom\(\infty \)groupoid between these objects, in \(L_W D\).
For \(C\) any category, there is a model structure \([C^{{\mathrm {op}}}, {\mathrm {sSet}}]_{{\mathrm {proj}}}\) on the category of simplicial presheaves over \(C\) (the projective model structure), whose weak equivalences and fibrations are those transformations that are objectwise so in \({\mathrm {sSet}}_{{\mathrm {Quillen}}}\). If \(C\) is equipped with the structure of a site given by a (pre)topology, then there are corresponding localizations of the simplicial presheaves. We are interested here in the case that \(C\) has enough points.
Definition 2.1
Notice here that, by definition of geometric morphism, the functor \(i^*\) is left adjoint to \(i_*\)—hence preserves all colimits—and in addition preserves all finite limits.
Example 2.2

The categories \({\mathrm {Mfd}}\) (\({\mathrm {SmoothMfd}}\)) of (smooth) finitedimensional, paracompact manifolds and smooth functions between them;

the category \({\mathrm {CartSp}}\) of Cartesian spaces \({\mathbb {R}}^n\) for \(n \in {\mathbb {N}}\) and continuous (smooth) functions between them.
These examples are discussed in more depth in [45]—we refer the reader there for further details. We restrict from now on attention to the case that \(C\) has enough points.
2.2 By simplicial objects in the site
Sometimes it is considered desirable to present an \(\infty \)stack by a simplicial presheaf which in turn is presented by a simplicial object in the underlying site. We observe here that this is always possible provided the site has arbitrary coproducts.
Definition 2.3
Example 2.4
Let \(C\) be a category of connected topological spaces with given extra structure and properties (for instance smooth manifolds). Then \({\bar{C}}\) is the category of all such spaces (with arbitrary many connected components).
Proposition 2.5
We will prove this shortly, after we have made the following observation.
Proposition 2.6
 1.
Every simplicial presheaf over \(C\) is equivalent in \([C^{{\mathrm {op}}}, {\mathrm {sSet}}]_{{\mathrm {proj}}}\) to a simplicial object in \({\bar{C}}\) under the image of the degreewise Yoneda embedding \(j : {\bar{C}}^{\Delta ^{{\mathrm {op}}}} \rightarrow [C^{{\mathrm {op}}}, {\mathrm {sSet}}] \).
 2.
If moreover \(C\) has pullbacks and sequential colimits, then the simplicial object in \({\bar{C}}\) can be taken to be globally Kan, hence fibrant in \([C^{{\mathrm {op}}}, {\mathrm {sSet}}]_{{\mathrm {proj}}}\).
This proposition can be interpreted as follows: every \(\infty \)stack over \(C\) has a presentation by a simplicial object in \({\bar{C}}\). Moreover this is true with respect to any Grothendieck topology on \(C\), since the weak equivalences in the global projective model structure remain weak equivalences in any left Bousfield localization. If moreover \(C\) has all pullbacks (for instance for topological spaces, but not for smooth manifolds) then every \(\infty \)stack over \(C\) even has a presentation by a globally Kan simplicial object in \({\bar{C}}\).
Proof
Proof of Proposition 2.5
Let \(Q : [C^{{\mathrm {op}}}, {\mathrm {sSet}}] \rightarrow {\bar{C}}^{\Delta ^{{\mathrm {op}}}}\) be Dugger’s functor from the proof of Proposition 2.6. In [14] it is shown that for all \(X\) the simplicial presheaf \(Q X\) is cofibrant in \([C^{{\mathrm {op}}}, {\mathrm {sSet}}]_{{\mathrm {proj}}}\) and that the natural morphism \(Q X \rightarrow X\) is a weak equivalence (as we have observed previously). Since left Bousfield localization does not affect the cofibrations and only enlarges the weak equivalences, the same is still true in \([C^{{\mathrm {op}}}, {\mathrm {sSet}}]_{{\mathrm {proj}}, {\mathrm {loc}}}\).
Remark 2.7
If the site \(C\) is moreover equipped with the structure of a geometry as in [39] then there is a canonical notion of a \(C\)manifold: a sheaf on \(C\) that is locally isomorphic to a representable in \(C\). Write \(C {\mathrm {Mfd}}\) for the full subcategory of the category of presheaves on the \(C\)manifolds.
Example 2.8
Remark 2.9
While the above gives fairly general conditions on a site \(C\) under which every \(\infty \)stack is presented by a simplicial object in the site, and in fact by a simplicial object which is cofibrant in the projective model structure on the simplicial presheaves over the site, this simplicial object is in general not fibrant in that model structure, nor will it be stalkwise fibrant in general.
In parts of the literature special attention is paid to \(\infty \)stacks (or just stacks) that admit a presentation by a simplicial presheaf which is both: 1. represented by a simplicial object in the site and 2. stalkwise Kan fibrant in a suitable sense. (For instance SchommerPries discusses this for 1stacks on manifolds and [58] (see there for further references) for \(\infty \)stacks on manifolds.) It is an interesting question—which is open at the time of this writing—what these conditions on the presentation of an \(\infty \)stack mean intrinsically, for instance if they can be interpreted as ensuring an abstract geometricity condition on an \(\infty \)stack, such as considered for instance in [39].
3 Presentation of structures in an \(\infty \)topos
In the companion article [44] we considered a list of structures present in any \(\infty \)topos, which form the fabric for our discussion of principal (and associated/twisted) \(\infty \)bundles. Here we go through the same list of notions and discuss aspects of their presentation in categories of simplicial (pre)sheaves.
3.1 Cones
Proposition 3.1

one of the two morphisms is a fibration and all three objects are fibrant;

one of the two morphisms is a fibration and the model structure is right proper.
This appears for instance as Proposition A.2.4.4 in [38].
Proposition 3.2
A finite homotopy limit computed in \([C^{{\mathrm {op}}}, {\mathrm {sSet}}]_{{\mathrm {proj}}}\) presents also the homotopy limit in \([C^{{\mathrm {op}}}, {\mathrm {sSet}}]_{{\mathrm {proj}},{\mathrm {loc}}}\).
Proposition 3.3
3.2 Effective epimorphisms
We discuss aspects of the presentation of effective epimorphisms in an \(\infty \)topos. We begin with the following observation.
Observation 3.4
If the \(\infty \)topos \({\mathbf {H}}\) is presented by a category of simplicial presheaves, Sect. 2.1, then for \(X\) a simplicial presheaf, the canonical morphism \({\mathrm {const}} X_0 \rightarrow X\) in \([C^{{{\mathrm {op}}}},{\mathrm {sSet}}]\) that includes the presheaf of 0cells as a simplicially constant simplicial presheaf presents an effective epimorphism in \({\mathbf {H}}\).
This follows with Proposition 7.2.1.14 in [38].
Remark 3.5
In practice the presentation of an \(\infty \)stack by a simplicial presheaf is often taken to be understood, and then Observation 3.4 induces also a canonical atlas, i.e. \({\mathrm {const}}X_0 \rightarrow X\).
Definition 3.6
This functor was introduced in [26]. A discussion in the present context can be found in section 2.2 of [51], amongst other references.
Proposition 3.7

the horizontal morphism is given in degree \(n\) by \(d_{n+1} : X_{n+1} \rightarrow X_n\);

the horizontal morphism is a Kan fibration if \(X\) is a Kan complex;

the vertical morphism is a simplicial deformation retraction, in particular a weak homotopy equivalence.
Proof
Corollary 3.8
For \(X\) in \([C^{{{\mathrm {op}}}}, {\mathrm {sSet}}]_{{\mathrm {proj}}}\) fibrant, a fibration resolution of the canonical effective epimorphism \({\mathrm {const}} X_0 \rightarrow X\) from Observation 3.4 is given by the décalage morphism \({\mathrm {Dec}}_0 X \rightarrow X\), Proposition 3.7.
Proof
3.3 Connected objects
In every \(\infty \)topos \({\mathbf {H}}\) there is a notion of connected objects, which form the objects of the full sub\(\infty \)category \({\mathbf {H}}_{\ge 1}\). We discuss here presentations of connected and of pointed connected objects in \({\mathbf {H}}\) by means of presheaves of pointed or reduced simplicial sets.
Observation 3.9
 1.
\(X\) is inhabited (not empty);
 2.
all simplicial homotopy groups \(\pi _k(X)\) of \(X\) in degree \(k \le n\) are trivial.
Definition 3.10
Proposition 3.11
Remark 3.12
For \((* \rightarrow X) \in {\mathrm {sSet}}^{*/}\) such that \(X \in {\mathrm {sSet}}\) is Kan fibrant and \(n\)connected, the counit \(E_{n+1}(X,*) \rightarrow X\) is a homotopy equivalence. This statement appears for instance as part of Theorem 8.4 in [42].
Proposition 3.13
Let \(C\) be a site with a terminal object and let \({\mathbf {H}} := {\mathrm {Sh}}_\infty (C)\). Then under the presentation \({\mathbf {H}} \simeq ([C^{{\mathrm {op}}}, {\mathrm {sSet}}]_{{\mathrm {proj}}, {\mathrm {loc}}})^\circ \) every pointed \(n\)connected object in \({\mathbf {H}}\) is presented by a presheaf of \(n\)reduced simplicial sets, under the canonical inclusion \([C^{{\mathrm {op}}}, {\mathrm {sSet}}_n] \hookrightarrow [C^{{\mathrm {op}}}, {\mathrm {sSet}}]\).
Proof
We next describe a slightly enhanced version of the model structure on reduced simplicial sets introduced by Quillen in [48].
Proposition 3.14
The category \({\mathrm {sSet}}_{0}\) of reduced simplicial sets carries a left proper combinatorial model category structure whose weak equivalences and cofibrations are those in \({\mathrm {sSet}}_{{\mathrm {Quillen}}}\) under the inclusion \({\mathrm {sSet}}_{0} \hookrightarrow {\mathrm {sSet}}\).
Proof
Lemma 3.15
A fibration \(f:X\rightarrow Y\) in \({\mathrm {sSet}}_{0}\) (for the model structure of Proposition 3.14) is a Kan fibration precisely if it has the right lifting property against the morphism \((* \rightarrow S^1) := {\mathrm {red}}(\Delta [0] \rightarrow \Delta [1])\). In particular every fibrant object in \({\mathrm {sSet}}_{0}\) is a Kan complex.
Proof
The first statement appears as V Lemma 6.6. in [23]. The second (an immediate consequence) as V Corollary 6.8.
Proposition 3.16
Proof
It is clear that the inclusion \(i:{\mathrm {sSet}}_0\hookrightarrow {\mathrm {sSet}}^{*/}_{{\mathrm {Quillen}}}\) preserves cofibrations and acyclic cofibrations, in fact all weak equivalences. Since the point is cofibrant in \({\mathrm {sSet}}_{{\mathrm {Quillen}}}\), the model structure on the right is by Proposition 3.3 indeed a presentation of \({\mathrm {Grpd}}_{\infty }^{*/}\).
3.4 Groupoids
In [4] a presentation of groupoid objects in \(\infty {\mathrm {Grpd}}\) is discussed in terms of simplicial objects in \({\mathrm {sSet}}_{{\mathrm {Quillen}}}\), called ‘invertible Segal spaces’ in [4]. This has a straightforward generalization to a presentation of groupoid objects in a sheaf \(\infty \)topos \({\mathrm {Sh}}_\infty (C)\) by simplicial objects in a category of simplicial presheaves. We discuss here a presentation of homotopy colimits over such simplical diagrams given by the diagonal simplicial set or the total simplicial set associated with a bisimplicial set. This serves as the basis for the discussion of universal weakly principal simplicial bundles below in Sect. 3.7.1. For some general background on homotopy colimits the way we need them here, a good survey is [21].
Proposition 3.17
 The simplex functoris a cofibrant resolution of \(*\) in \([\Delta , {\mathrm {sSet}}_{{\mathrm {Quillen}}}]_{\mathrm {Reedy}}\);$$\begin{aligned} \Delta : [n] \mapsto \Delta [n] := \Delta (,[n]) \end{aligned}$$
 the fat simplex functoris a cofibrant resolution of \(*\) in \([\Delta , {\mathrm {sSet}}_{{\mathrm {Quillen}}}]_{{\mathrm {proj}}}\).$$\begin{aligned} {\varvec{\Delta }} : [n] \mapsto N(\Delta /[n]) \end{aligned}$$
Proposition 3.18
 1.If every monomorphism in \(C\) is a cofibration, then the homotopy colimit over \(F\) is given by the realization, i.e.$$\begin{aligned} {\mathbb {L}}\mathrm{lim}_\rightarrow F \simeq \int \limits ^{[n] \in \Delta } F([n]) \cdot \Delta [n]. \end{aligned}$$
 2.If \(F\) takes values in cofibrant objects, then the homotopy colimit over \(F\) is given by the fat realization, i.e.$$\begin{aligned} {\mathbb {L}}\mathrm{lim}_\rightarrow F \simeq \int \limits ^{[n] \in \Delta } F([n]) \cdot {\varvec{\Delta }}[n]. \end{aligned}$$
 3.If \(F\) is Reedy cofibrant, then the canonical morphism(the Bousfield–Kan map) is a weak equivalence.$$\begin{aligned} \int \limits ^{[n] \in \Delta } F([n]) \cdot {\varvec{\Delta }}[n] \rightarrow \int \limits ^{[n] \in \Delta } F([n]) \cdot \Delta [n] \end{aligned}$$
Proposition 3.19
The homotopy colimit of a simplicial diagram in \({\mathrm {sSet}}_{{\mathrm {Quillen}}}\), or more generally of a simplicial diagram of simplicial presheaves, is given by the diagonal of the corresponding bisimplicial set/bisimplicial presheaf.
Proof
Definition 3.20

\({\mathrm {Dec}} := \sigma ^*\) is called the total décalage functor;

\(\sigma _*\) is called the total simplicial set functor.
The total simplicial set functor was introduced in [1], for further discussion see [11, 51].
Remark 3.21
Remark 3.22
For \(X \in [\Delta ^{{\mathrm {op}}}, {\mathrm {sSet}}]\), the simplicial set \(\sigma _*X\) is in each degree given by an equalizer of maps between finite products of components of \(X\) (see for instance equation (2) of [51]). Hence forming \(\sigma _*\) is compatible with sheafification and other processes that preserve finite limits.
Proposition 3.23
 for every \(X \in [\Delta ^{{\mathrm {op}}}, {\mathrm {sSet}}]\), the canonical morphismfrom the diagonal to the total simplicial set is a weak equivalence in \({\mathrm {sSet}}_{{\mathrm {Quillen}}}\);$$\begin{aligned} d X \rightarrow \sigma _* X \end{aligned}$$
 for every \(X\in {\mathrm {sSet}}\) the adjunction unitis a weak equivalence in \({\mathrm {sSet}}_{{\mathrm {Quillen}}}\).$$\begin{aligned} X \rightarrow \sigma _* \sigma ^* X \end{aligned}$$

there is a natural isomorphism \(\sigma _* {\mathrm {const}} X \simeq X\).
These statements are due to Cegarra and Remedios in [11] and independently Joyal and Tierney (unpublished)—see also [51].
Corollary 3.24
Proof
By Proposition 3.23 this follows from Proposition 3.19. \(\square \)
Remark 3.25
The use of the total simplicial set instead of the diagonal simplicial set in the presentation of simplicial homotopy colimits is useful and reduces to various traditional notions in particular in the context of group objects and action groupoid objects. We discuss this further in Sect. 3.5 and Sect. 3.7.1 below.
3.5 Groups
Every \(\infty \)topos \({\mathbf {H}}\) comes with a notion of \(\infty \)group object that generalizes the ordinary notion of group object in a topos as well as that of grouplike \(A_\infty \) space in \({\mathrm {Top}}\simeq {\mathrm {Grpd}}_{\infty }\). We discuss presentations of \(\infty \)group objects by presheaves of simplicial groups.
Definition 3.26
This simplicial delooping \(\overline{W}\) was originally introduced in [40]. The above formulation is due to Duskin, see Lemma 15 in [51].
Remark 3.27
The functor \(\overline{W}\) takes values in reduced simplicial sets, i.e. \(\overline{W}: [\Delta ^{{\mathrm {op}}},{\mathrm {Grp}}]\rightarrow {\mathrm {sSet}}_{{\mathrm {red}}}\).
Remark 3.28
For \(G\) a simplicial group, the simplicial set \(\overline{W}G\) is, by Corollary 3.24, the homotopy colimit over a simplicial diagram in simplicial sets. Below in Sect. 3.7.2 we see that this simplicial diagram is that presenting the groupoid object \(*/\!/G\) which is the action groupoid of \(G\) acting trivially on the point.
Proposition 3.29
The category \({\mathrm {sGrp}}\) of simplicial groups carries a cofibrantly generated model structure for which the fibrations and the weak equivalences are those of \({\mathrm {sSet}}_{{\mathrm {Quillen}}}\) under the forgetful functor \({\mathrm {sGrpd}} \rightarrow {\mathrm {sSet}}\).
Proof
This is originally due to [47], for a more recent account see V Theorem 2.3 in [23]. Note that since the model structure is therefore transferred along the forgetful functor, it inherits generating (acyclic) cofibrations from those of \({\mathrm {sSet}}_{{\mathrm {Quillen}}}\). \(\square \)
We now consider a presentation of the looping/delooping equivalence \({\mathrm {Grp}}({\mathbf {H}}) \simeq {\mathbf {H}}^{*/}_{\ge 1}\) due to Lurie, recalled as Theorem 2.14 in [44].
Theorem 3.30
 the adjunction unit is a weak equivalencefor every reduced simplicial set \(Y\),$$\begin{aligned} Y \mathop {\rightarrow }\limits ^{\simeq } \overline{W}L Y \end{aligned}$$

\(\overline{W}G\) is a Kan complex for any simplicial group \(G\).
This result is discussed for instance in chapter V of [23]; a new proof that the unit of the adjunction is a weak equivalence is given in [51].
Definition 3.31
This morphism is the standard presentation of the universal \(G\)principal simplicial bundle. We discuss this further in Sect. 3.7.1 below. The characterization by décalage of the total space \(W G\) is made fairly explicit on p. 85 of [16]; a fully explicit statement can be found in [49].
Proposition 3.32
The morphism \(W G \rightarrow \overline{W}G\) is a Kan fibration resolution of the point inclusion \({*} \rightarrow \overline{W}G\).
Proof
This follows directly from the characterization of \(W G \rightarrow \overline{W}G\) by décalage (Corollary 3.8). \(\square \)
This statement appears in [42] as the union of two results there: Lemma 18.2 of [42] gives the fibration property; Proposition 21.5 of [42] gives the contractibility of \(W G\).
Corollary 3.33
Proof
One finds that \(G\) is the 1categorical fiber of \(W G \rightarrow \overline{W}G\). The statement then follows using Proposition 3.32 together with Proposition 3.1. \(\square \)
The universality of \(W G \rightarrow \overline{W}G\) for \(G\)principal simplicial bundles is the topic of section 21 in [42].
Corollary 3.34
The Quillen equivalence \((L \dashv \overline{W})\) from Theorem 3.30 is a presentation of the looping/delooping equivalence \({\mathrm {Grp}}({\mathbf {H}}) \simeq {\mathbf {H}}^{*/}_{\ge 1}\) for the \(\infty \)topos \({\mathbf {H}} = {\mathrm {Grpd}}_{\infty }\).
We now lift all these statements from simplicial sets to simplicial presheaves.
Proposition 3.35
 every \(\infty \)group object has a presentation by a presheaf of simplicial groupswhich is fibrant in \([C^{{\mathrm {op}}}, {\mathrm {sSet}}]_{{\mathrm {proj}}}\);$$\begin{aligned} G \in [C^{{\mathrm {op}}}, {\mathrm {sGrp}}] \mathop {\rightarrow }\limits ^{U} [C^{{\mathrm {op}}}, {\mathrm {sSet}}] \end{aligned}$$
 the corresponding delooping object is presented by the presheafobtained from \(G\) by applying the functor \(\overline{W}\) objectwise.$$\begin{aligned} {\overline{W}} G \in [C^{{\mathrm {op}}}, {\mathrm {sSet}}_{0}] \hookrightarrow [C^{{\mathrm {op}}}, {\mathrm {sSet}}] \end{aligned}$$
Proof
Remark 3.36
We may read this as saying that every \(\infty \)group may be strictified.
3.6 Cohomology
We discuss presentations of the hom\(\infty \)groupoids, hence of cocycle \(\infty \)groupoids, hence of the cohomology in an \(\infty \)topos.

In Sect. 3.6.1 we study sufficient conditions on a simplicial presheaf \(A\) such that the ordinary simplicial hom \([C^{{\mathrm {op}}}, {\mathrm {sSet}}](Y,A)\) out of a split hypercover Open image in new window is already the correct derived hom out of \(X\). Since this simplicial hom is the Kan complex of simplicial hyperČech cocycles relative to \(Y\) with coefficients in \(A\), this may be taken to be a sufficient condition for \(A\)Čech cohomology to produce the correct intrinsic cohomology.

In Sect. 3.6.2 we consider not a full model category structure but just the structure of a category of fibrant objects. In this case there is no notion of split hypercover and instead one has to consider all possible covers and refinements between them. A central result of [7] shows that this produces the correct cohomology classes. Here we discuss the refinement of this classical statement to the full cocycle \(\infty \)groupoids.
3.6.1 By hyperČechcohomology in \(C\)acyclic simplicial groups
The condition on an object \(X \in [C^{{\mathrm {op}}}, {\mathrm {sSet}}]_{{\mathrm {proj}}}\) to be fibrant models the fact that \(X\) is an \(\infty \)presheaf of \(\infty \)groupoids. The condition that \(X\) is also fibrant as an object in \([C^{{\mathrm {op}}}, {\mathrm {sSet}}]_{{\mathrm {proj}},{\mathrm {loc}}}\) models the higher analog of the sheaf condition: it makes \(X\) an \(\infty \)sheaf/\(\infty \)stack. For generic sites, \(C\)fibrancy in the local model structure is a property rather hard to check or establish concretely. But often a given site can be replaced by another site on which the condition is easier to control, without changing the corresponding \(\infty \)topos, up to equivalence. Here we discuss a particularly nice class of sites called \(\infty \)cohesive sites [53], and describe explicit conditions for a simplicial presheaf over them to be fibrant.
Definition 3.37
 1.
it has a terminal object;
 2.there is a generating coverage such that for every generating cover \(\{U_i \rightarrow U\}\) we have
 (a)
the Čech nerve \(\check{C}(\{U_i\}) \in [C^{{\mathrm {op}}}, {\mathrm {sSet}}]\) is degreewise a coproduct of representables;
 (b)the limit and colimit functors, \(\varprojlim :[C^{{\mathrm {op}}},{\mathrm {sSet}}]\!\rightarrow \! {\mathrm {sSet}}\) and \(\mathrm{lim}_\rightarrow :[C^{{\mathrm {op}}}, {\mathrm {sSet}}] \!\rightarrow \! {\mathrm {sSet}}\) respectively, send the Čech nerve projection \(\check{C}(\{U_i\})\rightarrow U\) to a weak homotopy equivalence:and$$\begin{aligned} \mathrm{lim}_\rightarrow \check{C}(\{U_i\}) \xrightarrow {\simeq } \mathrm{lim}_\rightarrow U = * \end{aligned}$$$$\begin{aligned} \varprojlim \check{C}(\{U_i\}) \mathop {\rightarrow }\limits ^{\simeq } \varprojlim U. \end{aligned}$$
 (a)
Remark 3.38
This last condition is familiar from the nerve theorem [5]:
Theorem 3.39
Remark 3.40
The conditions on an \(\infty \)cohesive site ensure that the Čech nerve of a good cover is cofibrant in the projective model structure \([C^{{\mathrm {op}}}, {\mathrm {sSet}}]_{{\mathrm {proj}}}\) and hence also in its localization \([C^{{\mathrm {op}}}, {\mathrm {sSet}}]_{{\mathrm {proj}}, {\mathrm {loc}}}\).
In order to discuss descent over \(C\) it is convenient to introduce the following notation for ‘cohomology over the site \(C\)’. For the moment this is just an auxiliary technical notion. Later we will see how it relates to an intrinsically defined notion of cohomology.
Definition 3.41
Definition 3.42
Note that if \(X\) is a simplicial presheaf on \(C\), then \(\pi _n(X)\) is naturally a group object over the sheaf associated to \(X_0\). Using this we can state the main definition of this section.
Definition 3.43
 1.
it is fibrant in \([C^{{\mathrm {op}}}, {\mathrm {sSet}}]_{{\mathrm {proj}}}\);
 2.
for all \(n \in {\mathbb {N}}\) we have \(\pi _n^{{\mathrm {PSh}}}(A) = \pi _n(A)\), in other words the homotopy group presheaves from Definition 3.42 are already sheaves;
 3.the sheaves \(\pi ^{ PSh }_n(A)\) are acyclic with respect to good covers; i.e. for every object \(U\), for every point \(a_U\in A_0(U)\), and for all good covers \(\{U_i\rightarrow U\}\) of \(U\), we haveand$$\begin{aligned} H^{1}_C(\{U_i\},\pi _1(A,a_U)) = 1 \end{aligned}$$for all \(k \ge 1\) if \(n\ge 2\).$$\begin{aligned} H^k_C(\{U_i\},\pi _n(A,a_U)) = 1 \end{aligned}$$
Remark 3.44
This definition can be formulated and the following statements about it are true over any site whatsoever. However, on generic sites \(C\) the \(C\)acyclic objects are not very interesting. They become interesting on sites such as the \(\infty \)cohesive sites considered here, whose topology sees all their objects as being contractible.
Observation 3.45
If \(A\) is \(C\)acyclic then \(\Omega _x A\) is \(C\)acyclic for every point \(x : * \rightarrow A\) (for any model of the loop space object in \([C^{{\mathrm {op}}}, {\mathrm {sSet}}]_{{\mathrm {proj}}}\)).
Proof
Theorem 3.46

\(A\) is 0\(C\)truncated and \(C\)acyclic;

\(A\) is \(C\)connected and \(C\)acyclic;

\(A\) is a group object and \(C\)acyclic.
Here and in the following “\(C\)truncated” and “\(C\)connected” means: as simplicial presheaves (not after sheafification of homotopy presheaves). So for example, here and in the following a simplicial presheaf \(X\) is \(C\)connected if it takes values in connected simplicial sets.
Remark 3.47
This means that with \(A\) satisfying the conditions of Theorem 3.46 above, with \(X\) any simplicial presheaf and Open image in new window a split hypercover (see Definition 4.8 of [15]), the cocycle \(\infty \)groupoid \({\mathbf {H}}(X, A)\) is presented by simplicial function complex \([C^{{\mathrm {op}}}, {\mathrm {sSet}}](Y, A)\). The vertices of this simplicial set are simplicial hyperČech cocycles with coefficients in \(A\), the edges are Čech coboundaries and so on. Specifically, if \(\{U_i \rightarrow X\}\) is a good cover in that all finite nonempty intersections of patches are representable, then the Čech nerve \(\check{C}(\{U_i\}) \rightarrow X\) is a split hypercover, and a morphism of simplicial presheaves \(\check{C}(\{U_i\}) \rightarrow A\) is a hyperČech cocycle with respect to the given cover.
We demonstrate Theorem 3.46 in several stages in the following list of propositions.
Lemma 3.48
Proof
Lemma 3.49
 1.
\(H^0_C(U, A) \simeq *\);
 2.
\(\Omega _* A\) is fibrant in \([C^{{\mathrm {op}}}, {\mathrm {sSet}}]_{{\mathrm {proj}}, {\mathrm {loc}}}\),
Proof
 1.
\(\pi _0 {\mathrm {Maps}}(U, A) \rightarrow \pi _0{\mathrm {Maps}}(\check{C}(\{U_i\}), A)\)
 2.
\(\Omega _* {\mathrm {Maps}}(U, A) \rightarrow \Omega _* {\mathrm {Maps}}(\check{C}(\{U_i\}), A)\)
Lemma 3.50
An object \(A\) which is \(C\)connected, 1\(C\)truncated and \(C\)acyclic is fibrant in \([C^{{\mathrm {op}}}, {\mathrm {sSet}}]_{{\mathrm {proj}}, {\mathrm {loc}}}\).
Proof
The first condition of Lemma 3.49 holds by the third condition of \(C\)acyclicity. The second condition in Lemma 3.49 is that \(\pi _1(A)\) satisfies descent. By \(C\)acyclicity this is a sheaf and it is 0truncated by assumption, therefore it satisfies descent by Lemma 3.48. \(\square \)
Proposition 3.51
Every pointed \(C\)connected and \(C\)acyclic object \(A \in [C^{{\mathrm {op}}}, {\mathrm {sSet}}]_{{\mathrm {proj}}}\) is fibrant in \([C^{{\mathrm {op}}}, {\mathrm {sSet}}]_{{\mathrm {proj}}, {\mathrm {loc}}}\).
Proof
We first show the statement for truncated \(A\) and afterwards for the general case. The \(k\)truncated case in turn we consider by induction over \(k\). If \(A\) is 1truncated the proposition holds by Lemma 3.50. Assuming then that the statement has been shown for \(k\)truncated \(A\), we need to show it for \((k+1)\)truncated \(A\).
Moreover we see that \(K(n)\) is fibrant in \([C^{{\mathrm {op}}}, {\mathrm {sSet}}]_{{\mathrm {proj}}, {\mathrm {loc}}}\): the first condition of Sect. 3.49 holds by the assumption that \(A\) is \(C\)connected. The second condition is implied again by the induction hypothesis, since \(\Omega K(n)\) is \((n1)\)truncated, connected and still \(C\)acyclic, by Observation 3.45.
Lemma 3.52
Proof
Since homotopy pullbacks of presheaves are computed objectwise, it is sufficient to show this for \(C = *\), hence in \({\mathrm {sSet}}_{{\mathrm {Quillen}}}\). One checks that generally, for \(X\) a Kan complex and \(G\) a simplicial group acting on \(X\), the quotient morphism \(X \rightarrow X/G\) is a Kan fibration. Therefore the homotopy fiber of \(G \rightarrow G/G_0\) is presented by the ordinary fiber in \({\mathrm {sSet}}\). Since the action of \(G_0\) on \(G\) is free, this is indeed \(G_0 \rightarrow G\). \(\square \)
Proposition 3.53
Every \(C\)acyclic group object \(G \in [C^{{\mathrm {op}}}, {\mathrm {sSet}}]_{{\mathrm {proj}}}\) for which \(G_0\) is a sheaf is fibrant in \([C^{{\mathrm {op}}}, {\mathrm {sSet}}]_{{\mathrm {proj}}, {\mathrm {loc}}}\).
Proof
This completes the proof of Theorem 3.46.
3.6.2 By cocycles in a category of fibrant objects
We discuss here a presentation of the hom\(\infty \)groupoids of an \(\infty \)category which itself is presented by the homotopical structure known as a category of fibrant objects [7]. The resulting presentation is much ‘smaller’ than the general Dwyer–Kan simplicial localization [18]: where the latter encodes a morphism in the localization by a zigzag of arbitrary length (of morphisms in the presenting category), the following Theorem 3.61 asserts that with the structure of a category of fibrant objects, we may restrict to zigzags of length 1. A slight variant of this statement has been proven by Cisinski in [12]. The following subsumes this variant and provides a maybe more direct proof.
Before describing the hom\(\infty \)groupoids, we briefly recall some basic notions and facts from [7].
Definition 3.54
 1.
\({\mathcal C}_F\) and \({\mathcal C}_W\) contain all of the isomorphisms of \({\mathcal C}\),
 2.
weak equivalences satisfy the 2outof3 property,
 3.
the subcategories \({\mathcal C}_F\) and \({\mathcal C}_F\cap {\mathcal C}_W\) are stable under pullback,
 4.
there exist functorial path objects in \({\mathcal C}\).
The axioms for a category of fibrant objects give roughly half of the structure of a model category, however these axioms still suffice to give a calculusoffractions description of the associated homotopy category.
Example 3.55

For any model category (with functorial factorization) the full subcategory of fibrant objects is a category of fibrant objects.

The category of stalkwise Kan simplicial presheaves on any site with enough points. In this case the fibrations are the stalkwise fibrations and the weak equivalences are the stalkwise weak equivalences.
Remark 3.56
Notice that (over a nontrivial site) the second example above is not a special case of the first: while there are model structures on categories of simplicial presheaves whose weak equivalences are the stalkwise weak equivalences, their fibrations (even between fibrant objects) are much more restricted than just being stalkwise fibrations.
We will use repeatedly the following consequence of the axioms of a category of fibrant objects (this is called the cogluing lemma in [23] where it appears as Lemma 8.10, Chapter II).
Lemma 3.57
We now come to the discussion of the hom\(\infty \)groupoids presented by \({\mathcal C}\)
Definition 3.58
Let \({\mathcal C}\) be a category of fibrant objects and let \(X\) and \(A\) be objects of \({\mathcal C}\).
 objects are spans, hence diagrams in \({\mathcal C}\) of the form such that the left morphism is an acyclic fibration;
 morphisms \(f : (p_1,g_1) \rightarrow (p_2, g_2)\) are given by morphisms \(f : X \rightarrow Y\) in \({\mathcal C}\), making the diagram commute.
Remark 3.59
In Section 3.3 of [12] the category \({\mathrm {Cocycle}}(X,A)\) is denoted \(\underline{\mathrm {Hom}}_{\, {\mathcal C}}(X,A)\). In Section 1 of [29] the category \({\mathrm {wCocycle}}(X,A)\) (under different assumptions on \({\mathcal C}\)) is denoted \(H(X,A)\) (and there only the connected components are analyzed).
Remark 3.60
Theorem 3.61
As remarked above, a variant of this statement has been proven by Cisinski [12]—more precisely he has shown that the inclusion \(N{\mathrm {Cocycle}}(X,A) \rightarrow L^H{\mathcal C}(X,A)\) is a weak homotopy equivalence (see Proposition 3.23 of [12]]). We give here a direct proof of this result, which also establishes that \(N{\mathrm {Cocycle}}(X,A) \rightarrow N{\mathrm {wCocycle}}(X,A)\) is also a weak equivalence.
Let \(F^{1}{\mathcal C}^{i}(A,B)\) be the full subcategory of \(W^{1}{\mathcal C}^{i}(A,B)\) consisting of the zigzags where the left going morphism is as acyclic fibration rather than a weak equivalence. Analogously we write \(F^{1}{\mathcal C}^{i}F^{1}{\mathcal C}^{j}(A,B)\). Note that in either case the morphisms of these spans still consist of weak equivalences and not necessarily of acyclic fibrations.
Lemma 3.62
Proof
Lemma 3.63
Proof
Now the functor \(L_1\) restricts to a functor \(W^{1}W^{i}(A,B) \rightarrow F^{1}W^{i}(A,B)\) which is homotopy inverse to the second functor of the lemma.
Lemma 3.64
Each category of fibrant objects \({\mathcal C}\) admits a homotopy calculus of fractions and the chain of inclusions \(N{\mathrm {Cocycle}}(A,B) \rightarrow N{\mathrm {wCocycle}}(A,B) \rightarrow L^H{\mathcal C}(A,B)\) are all homotopy equivalences.
Proof
This completes the proof of Theorem 3.61.
3.7 Principal bundles
We discuss a presentation of the theory of principal \(\infty \)bundles from section 3 in [44].
3.7.1 Universal simplicial principal bundles and the Borel construction
By Proposition 3.35 every \(\infty \)group in an \(\infty \)topos over an \(\infty \)cohesive site is presented by a (pre)sheaf of simplicial groups, hence by a strict group object \(G\) in a 1category of simplicial (pre)sheaves. We have seen in Sect. 3.5 that, for such a presentation, the abstract delooping \({\mathbf {B}}G\) is presented by \(\overline{W}G\). By Theorem 3.19 in [44], the theory of \(G\)principal \(\infty \)bundles is essentially that of homotopy fibers of morphisms into \({\mathbf {B}}G\), and hence, for such a presentation, that of homotopy fibers of morphisms into \(\overline{W}G\). By Proposition 3.1 such homotopy fibers are computed as ordinary pullbacks of fibration resolutions of the point inclusion into \(\overline{W}G\). Here we discuss these fibration resolutions. They turn out to be the classical universal simplicial principal bundles \(W G \rightarrow \overline{W}G\) of Definition 3.31.
Let \(C\) be a site; we consider group objects in \([C^{{\mathrm {op}}}, {\mathrm {sSet}}]\). In the following let \(P \in [C^{{\mathrm {op}}}, {\mathrm {sSet}}]\) be an object equipped with an action \(\rho : P \times G \rightarrow P\) by a group object \(G\). Since sheafification preserves finite limits, all of the following statements hold verbatim also in the category \({\mathrm {sSh}}(C)\) of simplicial sheaves over \(C\).
Definition 3.65
Definition 3.66
Remark 3.67
According to Corollary 3.24 the object \(P/_h G\) presents the homotopy colimit over the simplicial object \(P/\!/G\). We say that \(P/_h G\) is the homotopy quotient of \(P\) by the action of \(G\).
Example 3.68
Definition 3.69
We will call this the universal weakly \(G\)principal bundle.
Remark 3.70
Traditionally, at least over the trivial site, this is known as a presentation of the universal \(G\)principal simplicial bundle; we review this traditional theory below in Sect. 4.1. However, when prolonged to presheaves of simplicial sets as considered here, it is not quite accurate to speak of a genuine universal principal bundle: because the pullbacks of this bundle to hypercovers will in general only be “weakly principal” in a sense that we discuss in a moment in Sect. 3.7.2. Therefore it is more accurate to speak of the universal weakly \(G\)principal bundle.
The following proposition (which appears as Lemma 10 in [49]) justifies this terminology and the notation \(WG\) (which, recall, has already been used in Definition 3.31).
Proposition 3.71
 1.
it is isomorphic to the décalage morphism \({\mathrm {Dec}}_0 \overline{W}G \rightarrow \overline{W}G\), Definition 3.31,
 2.
\(WG\) is canonically equipped with a right \(G\)action over \(\overline{W}G\) that makes \(WG\rightarrow \overline{W}G\) a \(G\)principal bundle.
In particular it follows from 2 that \(WG\rightarrow \overline{W}G\) is an objectwise Kan fibration replacement of the point inclusion \(*\rightarrow \overline{W}G\).
We now discuss some basic properties of the morphism \(WG\rightarrow \overline{W}G\).
Definition 3.72
Proposition 3.73
Proof
This follows by a straightforward computation. \(\square \)
Lemma 3.74
 1.
The quotient map \(P \rightarrow P/G\) is a Kan fibration.
 2.
The quotient \(P/G\) is a Kan complex.
The second statement is for instance Lemma 3.7 in Chapter V of [23].
Lemma 3.75
For \(P\) a Kan complex and \(P \times G \rightarrow P\) an action by a group object, the homotopy quotient \(P /_h G\), Definition 3.66, is itself a Kan complex.
Proof
By Proposition 3.73 the homotopy quotient is isomorphic to the Borel construction. Since \(G\) acts freely on \(WG\) it acts freely on \(P \times WG\). The statement then follows with Lemma 3.74.\(\square \)
Remark 3.76
Let \(\hat{X} \rightarrow \overline{W}G\) be a morphism in \([C^{{\mathrm {op}}}, {\mathrm {sSet}}]\), presenting, by Proposition 3.35, a morphism \(X \rightarrow {\mathbf {B}}G\) in the \(\infty \)topos \({\mathbf {H}} = {\mathrm {Sh}}_\infty (C)\). By theorem 3.19 of [44] every \(G\)principal \(\infty \)bundle over \(X\) arises as the homotopy fiber of such a morphism. By using Proposition 3.71 together with Proposition 3.1 it follows that the principal \(\infty \)bundle classified by \(\hat{X} \rightarrow \overline{W}G\) is presented by the ordinary pullback of \(W G \rightarrow \overline{W}G\). This is the defining property of the universal principal bundle.
In Sect. 3.7.2 below we show how this observation leads to a complete presentation of the theory of principal \(\infty \)bundles by weakly principal simplicial bundles.
3.7.2 Presentation in locally fibrant simplicial sheaves
We discuss a presentation of the general notion of principal \(\infty \)bundles, by weakly principal bundles in a 1category of simplicial sheaves.
Let \({\mathbf {H}}\) be a hypercomplete \(\infty \)topos (for instance a cohesive \(\infty \)topos), which admits a 1site \(C\) with enough points.
Observation 3.77
Corollary 3.78
Regard \({\mathrm {sSh}}(C)_{\mathrm {lfib}}\) as a category of fibrant objects, Definition 3.54, with weak equivalences and fibrations the stalkwise weak equivalences and fibrations in \({\mathrm {sSet}}_{{\mathrm {Quillen}}}\), respectively, as in Example 3.55. Then for any two objects \(X, A \in {\mathbf {H}}\) there are simplicial sheaves, to be denoted by the same symbols, such that the hom \(\infty \)groupoid in \({\mathbf {H}}\) from \(X\) to \(A\) is presented in \({\mathrm {sSet}}_{{\mathrm {Quillen}}}\) by the Kan complex of cocycles from Sect. 3.6.2.
Proof
By Theorem 3.61.\(\square \)
We now discuss, for the general theory of principal \(\infty \)bundles in \({\mathbf {H}}\) from [44] a corresponding realization in the presentation for \({\mathbf {H}}\) given by \(({\mathrm {sSh}}(C), W)\).
By Proposition 3.35 every \(\infty \)group in \({\mathbf {H}}\) is presented by an ordinary group in \({\mathrm {sSh}}(C)\). It is too much to ask that also every \(G\)principal \(\infty \)bundle is presented by a principal bundle in \({\mathrm {sSh}}(C)\). But something close is true: every principal \(\infty \)bundle is presented by a weakly principal bundle in \({\mathrm {sSh}}(C)\).
Definition 3.79

an object \(P \in {\mathrm {sSh}}(C)\) (the total space);

a local fibration \(\pi :P\rightarrow X\) (the bundle projection);
 the action of \(G\) is weakly principal in the sense that the shear mapis a local weak equivalence.$$\begin{aligned} (p_1, \rho ) : P \times G \rightarrow P \times _X P \quad (p,g) \mapsto (p,p g) \end{aligned}$$
Remark 3.80
We do not ask the \(G\)action to be degreewise free as in [31], where a similar notion is considered. However we show in Corollary 3.97 below that each weakly \(G\)principal bundle is equivalent to one with free \(G\)action.
Definition 3.81
Lemma 3.82
 1.for any point \(p : * \rightarrow P\) the action of \(G\) induces a weak equivalencewhere \(x = \pi (p)\) and where \(P_x\) is the fiber of \(P\rightarrow X\) over \(x\),$$\begin{aligned} G \longrightarrow P_x \end{aligned}$$
 2.for all \(n \in {\mathbb {N}}\), the multishear mapsare weak equivalences.$$\begin{aligned} P \times G^n \rightarrow P^{\times ^{n+1}_X} \qquad (p,g_1,\ldots ,g_n) \mapsto (p,p g_1,\ldots ,p g_n) \end{aligned}$$
Proof
We consider the first statement. Regard the weak equivalence \(P \times G \xrightarrow {\sim } P \times _X P\) as a morphism over \(P\) where in both cases the map to \(P\) is given by projection onto the first factor. By basic properties of categories of fibrant objects, both of these projections are fibrations. Therefore, by the cogluing lemma (Lemma 3.57) the pullback of the shear map along \(p\) is still a weak equivalence. But this pullback is just the map \(G\rightarrow P_x\), which proves the claim.
For the second statement, we use induction on \(n\). Suppose that \(P\times G^n\rightarrow P^{\times ^{n+1}_X}\) is a weak equivalence. By Lemma 3.57 again, the pullback \(P^{\times ^n_X}\times _X (P\times G)\rightarrow P^{\times ^{n+2}_X}\) of the shear map \(P\times G\rightarrow P\times _X P\) along the fibration \(P^{\times ^n_X} \rightarrow X\) is again a weak equivalence. Similarly the product \(P\times G^n\times G\rightarrow P^{\times ^{n+1}_X}\times G\) of the \(n\)fold shear map with \(G\) is also a weak equivalence. The composite of these two weak equivalences is the multishear map \( P \times G^{n+1} \rightarrow P^{\times ^{n+2}_X}\), which is hence also a weak equivalence. \(\square \)
Proposition 3.83
Proof
Again this follows by basic properties of a category of fibrant objects: the pullback \(f^*P\) exists and the morphism \(f^*P\rightarrow Y\) is again a local fibration; thus it only remains to show that \(f^*P\) is weakly principal, i.e. that the morphism \(f^*P \times G \rightarrow f^*P \times _Y f^*P\) is a weak equivalence. This follows from Lemma 3.57 again. \(\square \)
Remark 3.84
The next result says that weakly \(G\)principal bundles satisfy descent along local acyclic fibrations (hypercovers).
Proposition 3.85
Let \(p: Y \rightarrow X\) be a local acyclic fibration in \({\mathrm {sSh}}(C)\). Then the functor \(p_!\) defined above restricts to a functor \(p_!:{\mathrm {w}}G{\mathrm {Bund}}(Y) \rightarrow {\mathrm {w}}G{\mathrm {Bund}}(X)\), left adjoint to \(p^*:{\mathrm {w}}G{\mathrm {Bund}}(X) \rightarrow {\mathrm {w}}G{\mathrm {Bund}}(Y)\), hence to a homotopy equivalence in \({\mathrm {sSet}}_{{\mathrm {Quillen}}}\).
Proof
Given a weakly \(G\)principal bundle \(P \rightarrow Y\), the first thing we have to check is that the map \(P \times G \rightarrow P \times _X P\) is a weak equivalence. This map can be factored as \(P \times G \rightarrow P\times _Y P \rightarrow P \times _X P\). Hence it suffices to show that the map \(P \times _Y P \rightarrow P \times _X P\) is a weak equivalence. But this follows from Lemma 3.57, since both pullbacks are along local fibrations and \(Y \rightarrow X\) is a local weak equivalence by assumption. This establishes the existence of the functor \(p_!\). It is easy to see that it is left adjoint to \(p^*\). This implies that it induces a homotopy equivalence in \({\mathrm {sSet}}_{{\mathrm {Quillen}}}\). \(\square \)
Corollary 3.86
For \(f: Y \rightarrow X\) a local weak equivalence, the induced functor \(f^*: {\mathrm {w}}G{\mathrm {Bund}}(X) \rightarrow {\mathrm {w}}G{\mathrm {Bund}}(Y)\) is a homotopy equivalence.
Proof
Using the Factorization Lemma of [7] we can factor the weak equivalence \(f\) into a composite of a local acyclic fibration and a right inverse to a local acyclic fibration. Therefore, by Proposition 3.85, \(f^*\) may be factored as the composite of two homotopy equivalences, hence is itself a homotopy equivalence. \(\square \)
We discuss now how weakly \(G\)principal bundles arise from the universal \(G\)principal bundle (Definition 3.69) by pullback, and how this establishes their equivalence with \(G\)cocycles.
Proposition 3.87
For \(G\) a group object in \({\mathrm {sSh}}(C)\), the map \(W G \rightarrow \overline{W}G\) from Definition 3.69 equipped with the \(G\)action of Proposition 3.71 is a weakly \(G\)principal bundle.
Indeed, it is a genuine (strictly) \(G\)principal bundle, in that the shear map is an isomorphism. This is a classical fact, for instance around Lemma 4.1 in chapter V of [23]. In terms of the total simplicial set functor it is observed in Section 4 of [49].
Proof
Definition 3.88
Observation 3.89
Lemma 3.90
For \(p : Y \rightarrow X\) a local acyclic fibration, the morphism \(\sigma _* \check{C}(Y) \rightarrow X\) from Observation 3.89 is a local weak equivalence.
Proof
By pullback stability of local acyclic fibrations, for each \(n \in {\mathbb {N}}\) the morphism \(Y^{\times ^n_X} \rightarrow X\) is a local weak equivalence. By Remark 3.22 and Proposition 3.23 this degreewise local weak equivalence is preserved by the functor \(\sigma _*\). \(\square \)
The main statement now is the following.
Theorem 3.91
Proof
Lemma 3.92
Suppose that \(X\xrightarrow {p} Y \xrightarrow {q} Z\) is a diagram of simplicial sets such that \(p\) is a Kan fibration surjective on vertices and \(qp\) is a Kan fibration. Then \(q\) is also a Kan fibration.
This is Exercise V3.8 in [23].
We now discuss the equivalence between weakly \(G\)principal bundles and \(G\)cocycles. For \(X, A \in {\mathrm {sSh}}(C)\), write \({\mathrm {Cocycle}}(X,A)\) for the category of cocycles from \(X\) to \(A\), according to Sect. 3.6.
Definition 3.93
Observation 3.94
Theorem 3.95
Proof
Remark 3.96
Corollary 3.97
For each weakly \(G\)principal bundle \(P \rightarrow X\) there is a weakly \(G\)principal bundle \(P^{f}\) with a levelwise free \(G\)action and a weak equivalence \(P^{f} \xrightarrow \sim P\) of weakly \(G\)principal bundles over \(X\). In fact, the assignment \(P \mapsto P^f\) is an homotopy inverse to the full inclusion of weakly \(G\)principal bundles with free action into all weakly \(G\)principal bundles.
Proof
3.8 Associated bundles
In Section 4.1 of [44] is discussed a general notion of \(V\)fiber bundles which are associated to a \(G\)principal \(\infty \)bundle via an action of \(G\) on some \(V\). Here we discuss presentations of these structures in terms of the weakly principal simplicial bundles from Sect. 3.7.2.
In terms of this presentation, Proposition 4.6 in [44] has the following “strictification”.
Proposition 3.98
Let \(P \rightarrow X\) in \({\mathrm {sSh}}(C)_{\mathrm {lfib}}\) be a weakly \(G\)principal bundle with classifying cocycle Open image in new window according to Theorem 3.95. Then the \(\rho \)associated simplicial \(V\)bundle \(P \times _G V\) is locally weakly equivalent to the pullback of \({\mathbf {c}}\) along \(g\).
Proof
Remark 3.99
According to Theorem 4.11 in [44], every \(V\)fiber bundle in an \(\infty \)topos is associated to an \({\mathbf {Aut}}(V)\)principal \(\infty \)bundle. We observe that the main result of [56] is a presentation of this general theorem for 1localic \(\infty \)toposes (with a 1site of definition) in terms of simplicial presheaves.
4 Models

The trivial site models higher discrete geometry. We show how in this case the general theory reduces to the classical theory of ordinary simplicial principal bundles in Sect. 4.1.

The site of smooth manifolds models higher smooth geometry/differential geometry. Since this site does not have all pullbacks, item 2 of Proposition 2.6 does not apply, and so it is of interest to identify conditions under which a given principal \(\infty \)bundle is presentable not just by a simplicial smooth manifold, but by a locally Kan simplicial smooth manifolds. This we discuss in Sect. 4.2.
4.1 Discrete geometry
The terminal \(\infty \)topos is the \(\infty \)category \({\mathrm {Grpd}}_{\infty }\) of \(\infty \)groupoids, the one presented by the standard model category structures on simplicial sets and on topological spaces. Regarded as a gros \(\infty \)topos akin to that of smooth \(\infty \)groupoids discussed below in Sect. 4.2, we are to think of \({\mathrm {Grpd}}_{\infty }\) as describing discrete geometry: an object in \({\mathrm {Grpd}}_{\infty }\) is an \(\infty \)groupoid without extra geometric structure. In order to amplify this geometric perspective, we will sometimes speak of discrete \(\infty \)groupoids.
There is a traditional theory of strictly principal Kan simplicial bundles, i.e. simplicial bundles with \(G\) action for which the shear map is an isomorphism instead of, more generally, a weak equivalence, see also Remark 3.70. A classical reference for this is [42]. A standard modern reference is Chapter V of [23]. We now compare this classical theory of strictly principal simplicial bundles to the theory of weakly principal simplicial bundles according to Sect. 3.7.2.
Definition 4.1
 1.
the \(G\) action is degreewise free;
 2.
the canonical morphism \(P/G \rightarrow X\) out of the ordinary (1categorical) quotient is an isomorphism of simplicial sets.
In [23] this is Definitions 3.1 and 3.2 of Chapter V.
Lemma 4.2
Every morphism in \({\mathrm {s}}G{\mathrm {Bund}}(X)\) is an isomorphism.
In [23] this is Remark 3.3 of Chapter V.
Observation 4.3
Lemma 4.4
Every morphism of weakly principal simplicial bundles in \({\mathrm {KanCpx}}\) is a weak homotopy equivalence on the underlying Kan complexes.
Proposition 4.5
For \(G\) a simplicial group, the category \({\mathrm {sSet}}_G\) of \(G\)actions on simplicial sets and \(G\)equivariant morphisms carries the structure of a simplicial model category where the fibrations and weak equivalences are those of the underlying simplicial sets.
This is Theorem 2.3 of Chapter V in [23].
Corollary 4.6
For \(G\) a simplicial group and \(X\) a Kan complex, the slice category \({\mathrm {sSet}}_G/X\) carries a simplicial model structure where the fibrations and weak equivalences are those of the underlying simplicial sets, after forgetting the map to \(X\).
Lemma 4.7
Let \(G\) be a simplicial group and \(P \rightarrow X\) a weakly \(G\)principal simplicial bundle. Then the loop space \(\Omega _{(P \rightarrow X)} {\mathrm {Ex}}^\infty N ({\mathrm {w}}G{\mathrm {Bund}}(X))\) has the same homotopy type as the derived hom space \({\mathbb {R}}\mathrm {Hom}_{{\mathrm {sSet}}_G/X}(P,P)\).
Proof
By Theorem 2.3, Chapter V of [23] and Lemma 4.4 the free resolution \(P^f\) of \(P\) from Corollary 3.97 is a cofibrantfibrant resolution of \(P\) in the slice model structure of Corollary 4.6. Therefore the derived hom space is presented by the simplicial set of morphisms \({\mathrm {Hom}}_{{\mathrm {sSet}}_G/X}(P^f \cdot \Delta ^\bullet , P^f)\) and all these morphisms are equivalences. Therefore by Proposition 2.3 in [19] this simplicial set is equivalent to the loop space of the nerve of the subcategory of \({\mathrm {sSet}}_G/X\) on the weak equivalences connected to \(P^f\). By Lemma 4.4 this subcategory is equivalent (isomorphic even) to the connected component of \({\mathrm {w}}G{\mathrm {Bund}}(X)\) on \(P\). \(\square \)
Proposition 4.8

for all \(G\) and \(X\) an isomorphism on connected components;

not in general a weak equivalence in \({\mathrm {sSet}}_{{\mathrm {Quillen}}}\).
Proof
To see that the full embedding of strictly \(G\)principal bundles is also injective on connected components, notice that by Lemma 4.7 if a weakly \(G\)principal bundle \(P\) with degreewise free \(G\)action is connected by a zigzag of morphisms to some other weakly \(G\)principal bundle \(P\), then there is already a direct morphism \(P \rightarrow P'\). Since all strictly \(G\)principal bundles have free actions by definition, this shows that two of them that are connected in \({\mathrm {w}}G{\mathrm {Bund}}(X)\) are already connected in \({\mathrm {s}}G{\mathrm {Bund}}(X)\).
To see that in general \(N{\mathrm {s}}G{\mathrm {Bund}}(X)\) nevertheless does not have the correct homotopy type, it is sufficient to notice that the category \({\mathrm {s}}G{\mathrm {Bund}}(X)\) is always a groupoid, by Lemma 4.2. Therefore \(N{\mathrm {s}}G{\mathrm {Bund}}(X)\) is always a homotopy 1type. But by Theorem 3.95 the object \(N{\mathrm {w}}G{\mathrm {Bund}}(X)\) is not an \(n\)type if \(G\) is not an \((n1)\)type.
\(\square \)
Corollary 4.9
Proof
By Proposition 4.8 and Remark 3.96.\(\square \)
Remark 4.10
The first statement of Corollary 4.9 is the classical classification result for strictly principal simplicial bundles, for instance Theorem V3.9 in [23].
4.2 Smooth geometry
We discuss the canonical homotopy theoretic context for higher differential geometry.
Definition 4.11
Proposition 4.12
The inclusion \({\mathrm {CartSp}} \hookrightarrow {\mathrm {SmthMfd}}\) exhibits \({\mathrm {CartSp}}\) as a dense subsite of \({\mathrm {SmthMfd}}\). Accordingly, there is an equivalence of categories between the sheaf toposes over both sites, \({\mathrm {Sh}}({\mathrm {CartSp}}) \simeq {\mathrm {Sh}}({\mathrm {SmthMfd}})\).
Lemma 4.13
Definition 4.14
Proposition 4.15
 1.
It is hypercomplete.
 2.
It is equivalent to \({\mathrm {Sh}}_\infty ({\mathrm {SmthMfd}})\).
 3.
The site \(C\) is a \(\infty \)cohesive site (Definition 3.37).
These and the following statements are discussed in detail in Section 4.4. of [53]. In particular we have
Observation 4.16
Definition 4.17

a local weak equivalence if it is stalkwise a weak equivalence of simplicial sets;

a local fibration if it is stalkwise a Kan fibration of simplicial sets,
Proposition 4.18
Therefore the hom\(\infty \)groupoids are equivalently given by the cocycle categories of Proposition 3.61.
4.2.1 Locally fibrant simplicial manifolds
By Proposition 4.18 smooth \(\infty \)groupoids are presented by locally fibrant simplicial sheaves on \({\mathrm {CartSp}}\). Every simplicial manifold represents a simplicial sheaf over this site. We discuss now the full sub\(\infty \)category of \({\mathrm {Smooth}}{\mathrm {Grpd}}_{\infty }\) on those objects that are presented by locally Kan fibrant simplicial smooth manifolds.
Definition 4.19
The structure of a category of fibrant objects on \({\mathrm {sSh}}({\mathrm {CartSp}})_{\mathrm {lfib}}\), Proposition 4.18, does not quite transfer along this inclusion, because pullbacks in \({\mathrm {SmthMfd}}\) do not generally exist. Pullbacks in \({\mathrm {SmthMfd}}\) do however exist, notably, along surjective submersions.
Definition 4.20
 a submersive local fibration if \(f_0:X_0\rightarrow Y_0\) is a surjective submersion and for all \(0\le k\le n\), \(n\ge 1\) the canonical morphismis a surjective submersion;$$\begin{aligned} X_n\rightarrow Y_n\times _{M_{\Lambda ^k[n]}Y}M_{\Lambda ^k[n]}X \end{aligned}$$

a submersion if \(f_n : X_n \rightarrow Y_n\) is a submersion for each \(n \in {\mathbb {N}}\);

A simplicial smooth manifold \(X\) is said to be a Lie \(\infty \)groupoid if \(X \rightarrow *\) is a submersive local fibration and all of the face maps of \(X\) are submersions.
Example 4.21
Let \(X\) be a smooth manifold and \(\{U_i \rightarrow X\}\) an open cover. Then the Čech nerve projection \(\check{C}(\{U_i\}) \rightarrow X\) is a submersive local acyclic fibration between locally fibrant simplicial smooth manifolds.
Lemma 4.22
As a corollary we have the following statement.
Corollary 4.23
If \(f:X\rightarrow Y\) is a submersive local fibration, then \(f\) is a surjective submersion.
Proof
Proposition 4.24
The pullback of a (locally acyclic) submersive local fibration in \({\mathrm {sSmthMfd}}\) exists and is again a (locally acyclic) submersive local fibration.
Proof
To check the statement about local weak equivalences, use the facts that stalks commute with pullbacks and that acyclic fibrations in \( sSet \) are stable under pullback.
\(\square \)
4.2.2 Groups
By Theorem 3.30 every \(\infty \)group in \({\mathrm {Smooth}}{\mathrm {Grpd}}_{\infty }\) is presented by some group object in \({\mathrm {sSh}}({\mathrm {CartSp}})\). In view of the discussion in Sect. 4.2.1 it is of interest to determine those which are in the inclusion \({\mathrm {sSmthMfd}}_{\mathrm {lfib}} \hookrightarrow {\mathrm {sSh}}({\mathrm {CartSp}})\) from Definition 4.19.
Proposition 4.25
Let \(G\) be a simplicial Lie group. Then \(G\) is a Lie \(\infty \)groupoid, and so in particular is a locally fibrant simplicial smooth manifold, Definition 4.19.
Proof
When \(n=1\) we need to show that the two face maps \(d_0,d_1:G_1\rightarrow G_0\) are surjective submersions which is again clear since \(s_0:G_0\rightarrow G_1\) is a global section of both of these maps. When \(n=2\) the matching objects \(M_{\Lambda ^2_k}G\) for \(0\le k\le 2\) can be identified with pullbacks \(G_1\times _{G_0}G_1\) which exist in \({\mathrm {SmthMfd}}\) since \(d_0,d_1:G_1\rightarrow G_0\) are submersions. The Yoneda argument above then shows that \(G_2\rightarrow M_{\Lambda ^2_k}G\) is a surjective submersion in these cases.
This observation forms the basis for a proof by induction on \(n\ge 1\) that for any simplicial Lie group \(G\), and any integer \(0\le k\le n\), the limit \(M_{\Lambda ^n_k}G\) exists in \({\mathrm {SmthMfd}}\) (the Yoneda argument above then shows that \(G\) is locally fibrant). The case \(n=1\) is clear.
For the case \(k=n\) we apply the statement just proven with \(G\) replaced by its opposite simplicial Lie group \(G^o\); this has the property that \(M_{\Lambda ^n_0}G^o = M_{\Lambda ^n_{n}}G\), which shows that the limit \(M_{\Lambda ^n_n}G\) exists, completing the inductive step. \(\square \)
4.2.3 Principal bundles
By the discussion in Sect. 3.7.2 and using Proposition 4.18 we have a presentation of principal \(\infty \)bundles in the \(\infty \)topos \({\mathrm {Smooth}}{\mathrm {Grpd}}_{\infty }\) by weakly principal bundles in the category \({\mathrm {sSh}}({\mathrm {CartSp}})_{\mathrm {lfib}}\) of locally fibrant simplicial sheaves. Here we discuss how parts of this construction may be restricted further along the inclusion \({\mathrm {sSmthMfd}}_{\mathrm {lfib}} \hookrightarrow {\mathrm {sSh}}({\mathrm {CartSp}})_{\mathrm {locfib}}\) of locally fibrant simplicial smooth manifolds, Sect. 4.2.1
Proposition 4.26
 1.
the object \(\overline{W}G \in {\mathrm {sSh}}({\mathrm {CartSp}})\), Definition 3.26, is presented by a submersively locally fibrant simplicial smooth manifold.
 2.
the universal \(G\)principal bundle \(W G \rightarrow \overline{W}G\), Definition 3.31, formed in \({\mathrm {sSh}}({\mathrm {CartSp}})\) is presented by a submersive local fibration of simplicial smooth manifolds.
Proof
We first prove 1. Our proof of this essentially follow the proof of the corresponding result (Lemma 4.3) in [51], some extra care is needed however since it is not immediately clear that all of the requisite limits exist in \({\mathrm {SmthMfd}}\). Therefore we will prove by induction on \(n\ge 1\) that for any simplicial Lie group \(G\) and any integer \(0\le k\le n\), the limit \(M_{\Lambda ^n_k}\overline{W}G\) exists in \({\mathrm {SmthMfd}}\) and the canonical map \(\overline{W}G_n\rightarrow M_{\Lambda ^n_k}\overline{W}G\) is a surjective submersion.
 (a)
the limit \(M_{\Lambda ^{n1}_{k1}}WG\) exists in \({\mathrm {SmthMfd}}\),
 (b)the mapis a surjective submersion$$\begin{aligned} WG_{n1}\rightarrow M_{\Lambda ^{n1}_{k1}}WG\times _{M_{\Lambda ^{n1}_{k1}}\overline{W}G}\overline{W}G_{n1} \end{aligned}$$
To complete the inductive step we need to deal with the case when \(k=n\). Just as in the proof of Proposition 4.25 above, we can settle this case by replacing the group \(G\) with its opposite simplicial group \(G^o\).
It remains to prove the statements (a) and (b) above and the second statement of the Proposition. Before we do so, let us note that in analogy with Definition 4.1 we have a notion of a strictly principal bundle in simplicial manifolds, the only difference being that we require the bundle projection to be a submersion.
Definition 4.27
Let \(G\) be a simplicial Lie group and let \(X\) be a simplicial manifold. A strictly principal \(G\)bundle on \(X\) is a simplicial manifold \(P\) together with a submersion \(P\rightarrow X\) and an action of \(G\) on \(P\) such that for every \(n\ge 0\), the action of \(G_n\) on \(P_n\) equips \(P_n\rightarrow X_n\) with the structure of a (smooth) principal \(G_n\) bundle.
To prove the second statement of the Proposition, and the statements (a) and (b) above, it is enough to prove the following lemmas.
Lemma 4.28
Lemma 4.29
Proof of Lemma 4.28
Proof of Lemma 4.29
The limit \(M_{\Lambda ^n_k}P\), if it exists, is uniquely determined by the requirement that \(M_{\Lambda ^n_k}P\rightarrow M_{\Lambda ^n_k}X\) is a smooth \(M_{\Lambda ^n_k}G\) bundle, and that \(P_n\rightarrow M_{\Lambda ^n_k}P\) is equivariant for the homomorphism \(h^n_k:G_n\rightarrow M_{\Lambda ^n_k}G\). Since the quotient \(P_n/\ker (h^n_k)\) of \(P_n\) by the free action of the normal Lie subgroup \(\ker (h^n_k)\) has both of these properties, it follows that \(M_{\Lambda ^n_k}P\) exists and is isomorphic to \(P_n/\ker (h^n_k)\).\(\square \)
Proposition 4.30
Let \(G\) be a simplicial Lie group which presents a smooth \(\infty \)group in \({\mathrm {Grp}}({\mathrm {Smooth}}{\mathrm {Grpd}}_{\infty })\). Suppose that \(\overline{W}G\) is \({\mathrm {CartSp}}\)acyclic (Definition 3.43). Then every \(G\)principal \(\infty \)bundle over a smooth manifold \(X \in {\mathrm {SmthMfd}} \hookrightarrow {\mathrm {Smooth}}{\mathrm {Grpd}}_{\infty }\) has a presentation by a weakly principal \(G\)bundle \(P\rightarrow X\) for which \(P\) is a locally fibrant simplicial smooth manifold and \(P\rightarrow X\) is a submersive local fibration.
Proof
Notes
Acknowledgments
The writeup of this article and the companions [44, 45] was initiated during a visit by the first two authors to the third author’s institution, University of Glasgow, in summer 2011. It was completed in summer 2012 when all three authors were guests at the Erwin Schrödinger Institute in Vienna. The authors gratefully acknowledge the support of the Engineering and Physical Sciences Research Council grant number EP/I010610/1 and the support of the ESI; DS gratefully acknowledges the support of the Australian Research Council (Grant Number DP120100106).
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