Abstract
We study the homotopy theory of \(\infty \)categories enriched in the \(\infty \)category of simplicial spaces. That is, we consider enriched \(\infty \)categories as presentations of ordinary \(\infty \)categories by means of a “local” geometric realization functor , and we prove that their homotopy theory presents the \(\infty \)category of \(\infty \)categories, i.e. that this functor induces an equivalence from a localization of the \(\infty \)category of enriched \(\infty \)categories. Following Dwyer–Kan, we define a hammock localization functor from relative \(\infty \)categories to enriched \(\infty \)categories, thus providing a rich source of examples of enriched \(\infty \)categories. Simultaneously unpacking and generalizing one of their key results, we prove that given a relative \(\infty \)category admitting a homotopical threearrow calculus, one can explicitly describe the homspaces in the \(\infty \)category presented by its hammock localization in a much more explicit and accessible way. As an application of this framework, we give sufficient conditions for the Rezk nerve of a relative \(\infty \)category to be a (complete) Segal space, generalizing joint work with Low.
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1 Introduction
1.1 Introducing (even more) homotopy theory
In their groundbreaking papers [1, 2], Dwyer–Kan gave the first presentation of the \(\infty \)category of \(\infty \)categories, namely the category of categories enriched in simplicial sets: in modern language, every enriched category has an underlying \(\infty \)category, and this association induces an equivalence
from the (\(\infty \)categorical) localization of the category at the subcategory of Dwyer–Kan weak equivalences to the \(\infty \)category of \(\infty \)categories. Moreover, Dwyer–Kan provided a method of “introducing homotopy theory” into a category equipped with a subcategory of weak equivalences, namely their hammock localization functor of [1].
In this paper, we set up an analogous framework in the setting of \(\infty \)categories: we prove that the \(\infty \)category of \(\infty \)categories enriched in simplicial spaces likewise models the \(\infty \)category of \(\infty \)categories via an equivalence
and we define a hammock localization functor which likewise provides a method of “introducing (even more) homotopy theory” into relative \(\infty \)categories. We moreover prove the following two results – the first generalizing a theorem of Dwyer–Kan, the second generalizing joint work with Low (see [5]).
Theorem
(4.4). Given a relative \(\infty \)category admitting a homotopical threearrow calculus, the homspaces in the underlying \(\infty \)category of its hammock localization admit a canonical equivalence
from the groupoid completion of the \(\infty \)category of threearrow zigzags in .
Theorem
(6.1). Given a relative \(\infty \)category , its Rezk nerve

is a Segal space if admits a homotopical threearrow calculus, and

is moreover a complete Segal space if moreover is saturated and satisfies the twooutofthree property.
(The notion of a homotopical threearrow calculus is a minor variant on Dwyer–Kan’s “homotopy calculus of fractions” (see Definition 4.1). Meanwhile, the Rezk nerve is a straightforward generalization of Rezk’s “classification diagram” construction, which we introduced in [11] and proved computes the \(\infty \)categorical localization (see [11, Theorem 3.8 and Corollary 3.12]).)
Remark 1.1
In Remark 2.21, we show how our notion of “enriched \(\infty \)category” fits with the corresponding notion coming from Lurie’s theory of distributors.
Remark 1.2
Many of the original Dwyer–Kan definitions and proofs are quite pointset in nature. However, when working \(\infty \)categorically, it is essentially impossible to make such ad hoc constructions. Thus, we have no choice but to be both much more careful and much more precise in our generalization of their work.^{Footnote 1} We find Dwyer–Kan’s facility with universal constructions (displayed in that proof and elsewhere) to be really quite impressive, and we hope that our elaboration on their techniques will be pedagogically useful. Broadly speaking, our main technique is to corepresent higher coherence data.
1.2 Conventions
Though it stands alone, this paper belongs to a series on model \(\infty \)categories. These papers share many key ideas; thus, rather than have the same results appear repeatedly in multiple places, we have chosen to liberally crossreference between them. To this end, we introduce the following “code names”.
Title  Reference  Code 

Model \(\infty \)categories I: some pleasant properties of the \(\infty \)category of simplicial spaces  [10]  S 
The universality of the Rezk nerve  [11]  N 
All about the Grothendieck construction  [12]  G 
Hammocks and fractions in relative \(\infty \)categories  n/a  H 
Model \(\infty \)categories II: Quillen adjunctions  [13]  Q 
Model \(\infty \)categories III: the fundamental theorem  [14]  M 
Thus, for instance, to refer to [10, Theorem 1.9], we will simply write Theorem M.1.9. (The letters are meant to be mnemonical: they stand for “simplicial space”, “nerve”, “Grothendieck”, “hammock”, “Quillen”, and “model”, respectively.)
We take quasicategories as our preferred model for \(\infty \)categories, and in general we adhere to the notation and terminology of [7, 9]. In fact, our references to these two works will be frequent enough that it will be convenient for us to adopt Lurie’s convention and use the code names T and A for them, respectively.
However, we work invariantly to the greatest possible extent: that is, we primarily work within the \(\infty \)category of \(\infty \)categories. Thus, for instance, we will omit all technical uses of the word “essential”, e.g. we will use the term unique in situations where one might otherwise say “essentially unique” (i.e. parametrized by a contractible space). For a full treatment of this philosophy as well as a complete elaboration of our conventions, we refer the interested reader to §S.A. The casual reader should feel free to skip this on a first reading; on the other hand, the careful reader may find it useful to peruse that section before reading the present paper. For the reader’s convenience, we also provide a complete index of the notation that is used throughout this sequence of papers in §S.B.
1.3 Outline
We now provide a more detailed outline of the contents of this paper.

In Sect. 2, we introduce the \(\infty \)category of \(\infty \)categories enriched in simplicial spaces, as well as an auxiliary \(\infty \)category of Segal simplicial spaces. We endow both of these with subcategories of Dwyer–Kan weak equivalences, and prove that the resulting relative \(\infty \)categories both model the \(\infty \)category of \(\infty \)categories.

In Sect. 3, we define the \(\infty \)categories of zigzags in a relative \(\infty \)category between two objects , and use these to define the hammock simplicial spaces , which will be the homsimplicial spaces in the hammock localization .

In Sect. 4, we define what it means for a relative \(\infty \)category to admit a homotopical three arrow calculus, and we prove the first of the two results stated above.

In Sect. 5, we finally construct the hammock localization functor on relative \(\infty \)categories, and we explore some of its basic features.

In Sect. 6, we prove the second of the two results stated above.
2 Segal spaces, Segal simplicial spaces, and enriched \(\infty \)categories
In this section, we develop the theory—and the homotopy theory—of two closely related flavors of higher categories whose homobjects lie in the symmetric monoidal \(\infty \)category of simplicial spaces equipped with the cartesian symmetric monoidal structure. By “homotopy theory”, we mean that we will endow the \(\infty \)categories of these objects with relative \(\infty \)category structures, whose weak equivalences are created by “local” (i.e. homobjectwise) geometric realization. These therefore constitute “manyobject” elaborations on the Kan–Quillen relative \(\infty \)category , whose weak equivalences are created by geometric realization (see Theorem S.4.4). A key source of such objects will be the hammock localization functor, which we will introduce in Sect. 5.
This section is organized as follows.

In Sect. 2.1, we recall some basic facts regarding Segal spaces.

in Sect. 2.2, we introduce Segal simplicial spaces and define the essential notions for “doing (higher) category theory” with them.

In Sect. 2.3, we introduce their full (in fact, coreflective) subcategory of simpliciospatiallyenriched (or simply enriched) \(\infty \)categories. These are useful since they can more directly be considered as “presentations of \(\infty \)categories”.

In Sect. 2.4, we prove that freely inverting the Dwyer–Kan weak equivalences among either the Segal simplicial spaces or the enriched \(\infty \)categories yields an \(\infty \)category which is canonically equivalent to itself. We also contextualize both of these sorts of objects with respect to the theory of enriched \(\infty \)categories based in the notion of a distributor, and provide some justification for our interest in them.
2.1 Segal spaces
We begin this section with the following recollections. This subsection exists mainly in order to set the stage for the remainder of the section; we refer the reader seeking a more thorough discussion either to the original paper [16] (which uses model categories) or to [8, §1] (which uses \(\infty \)categories).
Definition 2.1
The \(\infty \)category of Segal spaces is the full subcategory of those simplicial spaces satisfying the Segal condition. These sit in a left localization adjunction
which factors the left localization adjunction of Definition N.2.1 in the sense that we obtain a pair of composable left localization adjunctions
(This follows easily from [16, Theorems 7.1 and 7.2], or alternatively moreorless follows from [8, Remark 1.2.11].)
In order to make a few basic observations, it will be convenient to first introduce the following.
Definition 2.2
Suppose that admits finite products. Then, we define the \(\mathbf{0}\)th coskeleton of an object (or perhaps more standardly, of the corresponding constant simplicial object ) to be the simplicial object selected by the composite
This assembles to a functor
which, as the notation suggests, is given in degree n by \(c \mapsto c^{\times (n+1)}\). This sits in an adjunction
which we refer to as the \(\mathbf{0}\)th coskeleton adjunction for . Using this, given a simplicial object and a map in , we define the pullback of Z along \(\varphi \) to be the fiber product
in , where the vertical map is the component at the object of the unit of the 0th coskeleton adjunction. In particular, note that we have a canonical equivalence \((\varphi ^*(Z))_0 \simeq Y\) in .
Remark 2.3
Suppose that , and let us write for its localization map. Then, the map is a surjection in , and moreover we have a canonical equivalence
in . (The first claim follows from [16, Theorem 7.7 and Corollary 6.5], while the second claim follows from combining [8, Definition 1.2.12(b) and Theorem 1.2.13(2)] with the Segal condition for .) From here, it follows easily that we have an equivalence
where denotes the full subcategory on those functors that select surjective functors . From this viewpoint, the left localization is then just the composite functor
where denotes the \(\infty \)categorical nerve functor. Thus, one might think of as “the \(\infty \)category of surjectively marked \(\infty \)categories” (where by “surjectively marked” we mean “equipped with a surjective map from an \(\infty \)groupoid”).
Remark 2.4
Continuing with the observations of Remark 2.3, note that the category of strict 1categories can be recovered as a limit
in (in which the square is already a pullback). (In fact, the inclusion itself fits into the defining pullback square
in .) We can therefore consider the \(\infty \)category of Segal spaces as a close cousin of the 1category of strict categories, with the caveat that objects of must be surjectively marked by a discrete space.
Remark 2.5
Suppose that . Then, we can compute homspaces in the \(\infty \)category
as follows. Any pair of objects can be considered as defining a pair of points
Since the map is a surjection, these admit lifts \(\tilde{x},\tilde{y} \in Y_0\). Then, we have a composite equivalence
by Remarks N.2.2 and 2.3. (In particular, we can compute the homspace using any choices of lifts \(\tilde{x},\tilde{y} \in Y_0\).)
2.2 Segal simplicial spaces
We now turn from the enriched context to the enriched context.
Definition 2.6
We define the \(\infty \)category of Segal simplicial spaces to be the full subcategory of those simplicial objects in which satisfy the Segal condition. These sit in a left localization adjunction by the adjoint functor theorem (Corollary T.5.5.2.9). We take the convention that our bisimplicial spaces are organized according to the diagram
in : we think of the columns as the “internal” simplicial spaces, and denote them as (omitting the outer index if it’s irrelevant for the discussion). The Segal condition then asserts that the map
is an equivalence in .
Remark 2.7
In light of Remark 2.4, we can consider the \(\infty \)category of Segal simplicial spaces as being a homotopical analog of the 1category of simplicial objects in strict 1categories. The subcategory of enriched categories then corresponds to the full subcategory on those Segal simplicial spaces such that the object is constant. We will restrict our attention to such objects in Sect. 2.3.
Definition 2.8
For any , we define the space of objects of to be the space
and for any , we define the homsimplicial space from x to y in to be the pullback
in . We refer to the points of the space
simply as morphisms from x to y. The various homsimplicial spaces of admit associative composition maps
in , which are obtained as usual via the Segal conditions. For any there is an evident identity morphism from x to itself, denoted , which behaves as expected under these composition maps.
Definition 2.9
Given any and any pair of objects , we say that two morphisms
are simplicially homotopic if the induced maps
are equivalent (i.e. select points in the same path component of the target). We then say that a morphism is a simplicial homotopy equivalence if there exists a morphism such that the composite morphisms
and
are simplicially homotopic to the respective identity morphisms.
Now, the objects of will indeed be “presentations of \(\infty \)categories”, but maps between them which are not equivalences may nevertheless induce equivalences between the \(\infty \)categories that they present. We therefore introduce the following notion.
Definition 2.10
A map in is called a Dwyer–Kan weak equivalence if

it is weakly fully faithful, i.e. for all pairs of objects the induced map
is an equivalence in \(\mathcal {S}\), and

it is weakly surjective, i.e. the map
is surjective up to the equivalence relation on generated by simplicial homotopy equivalence.
Such morphisms define a subcategory containing all the equivalences and satisfying the twooutofthree property, and we denote the resulting relative \(\infty \)category by .
Remark 2.11
Via the evident functor (recall Remark 2.7), the subcategory of Dwyer–Kan weak equivalences of Sect. 1.1 (i.e. the subcategory of weak equivalences for the Bergner model structure) is pulled back from the subcategory .
2.3 enriched \(\infty \)categories
In light of the discussion of Sect. 2.2, the natural guess for the sense in which a Segal simplicial space should be considered as a “presentation of an \(\infty \)category” is via the levelwise geometric realization functor
However, this operation does not preserve Segal objects: taking fiber products of simplicial spaces does not generally commute with taking their geometric realizations. On the other hand, these two operations do commute when the common target of the cospan is constant. Hence, it will be convenient to restrict our attention to the following special class of objects.
Definition 2.12
We define the \(\infty \)category of simpliciospatiallyenriched \(\infty \)categories, or simply of enriched \(\infty \)categories, to be the full subcategory on those objects such that is constant. We write
for the defining inclusion. Restricting the subcategory of Dwyer–Kan weak equivalences along this inclusion, we obtain a relative \(\infty \)category (which also has the twooutofthree property).
Lemma 2.13
There is a canonical factorization
of the restriction of the levelwise geometric realization functor
to the subcategory of enriched \(\infty \)categories.
Proof
Choose any . Since the functor is the inclusion of a full subcategory, it suffices to show that , for which in turn it suffices to show that the evident map
is an equivalence. Towards this aim, write
for the decomposition of into its connected components; since by assumption , this induces a decomposition
of . \(\mathcal {C}_1 \simeq \coprod _i (\mathcal {C}_1)_i\) and \(\mathcal {C}_{n1} \simeq \coprod _i (\mathcal {C}_{n1})_i\) for the resulting pulled back decompositions. Then, using Lemma A.5.5.6.17 (applied to the \(\infty \)topos \(\mathcal {S}\)) and the fact that coproducts commute with connected limits, we can identify the target of the above map as
As satisfies the Segal condition by assumption, this proves the claim. \(\square \)
Remark 2.14
The proof of Lemma 2.13 shows that it would suffice to make the weaker assumption that the object is constant in order to conclude that .
Definition 2.15
We denote simply by
the factorization of Lemma 2.13, and refer to it as the geometric realization functor on enriched \(\infty \)categories.
Definition 2.16
The composite inclusion
clearly factors through the subcategory . We simply write
for this factorization, and refer to it as the constant enriched \(\infty \)category functor. Thus, for an \(\infty \)category , the simplicial object
is given in degree n by
the constant simplicial space on the object
This functor clearly participates in a commutative diagram
in .
Remark 2.17
Suppose we are given a Segal simplicial space and a map in to its space of objects. Write for the corresponding map in . Then, the canonical map
is fully faithful (in the enriched sense): for any objects , the induced map
is already an equivalence in (instead of just being an equivalence upon geometric realization). Of course, the map is therefore in particular weakly fully faithful as well. As we can always choose our original map so that the induced map is additionally weakly surjective (e.g. by taking \(\varphi \) to be a surjection), it follows that any Segal simplicial space admits a Dwyer–Kan weak equivalence from a enriched category; indeed, we can even arrange to have .
Improving on Remark 2.17, we now describe a universal way of extracting a enriched \(\infty \)category from a Segal simplicial space.
Definition 2.18
We define the spatialization functor as follows.^{Footnote 2} Any gives rise to a natural map
in , the component at of the counit of the right localization adjunction . The spatialization of is then the pullback
(Note that the fiber product of Definition 2.2 that yields this pullback may be equivalently taken either in or in , in light of the left localization adjunction of Definition 2.6.) This clearly assembles to a functor, and in fact it is not hard to see that this participates in a right localization adjunction
whose counit components are Dwyer–Kan weak equivalences (which are even fully faithful as in Remark 2.17).
2.4 and as presentations of
The following pair of results asserts that both enriched \(\infty \)categories and Segal simplicial spaces, equipped with their respective subcategories of Dwyer–Kan weak equivalences, present the \(\infty \)category of \(\infty \)categories.
Proposition 2.19
The composite functor
induces an equivalence
Proof
So far, we have obtained the solid diagram
The right adjoint of the composite left localization adjunction
clearly lands in the full subcategory , and hence restricts to give the right adjoint of a left localization adjunction as indicated by the dotted arrow above. This composes to a left localization adjunction
Moreover, the definition of Dwyer–Kan weak equivalence is precisely chosen so that the composite left adjoint creates the subcategory [i.e. it is pulled back from the subcategory of equivalences (see Definition N.1.5)]. Hence, by Example N.1.13, it does indeed induce an equivalence
as desired. \(\square \)
Proposition 2.20
Both adjoints in the right localization adjunction
are functors of relative \(\infty \)categories (with respect to their respective Dwyer–Kan relative structures), and moreover they induce inverse equivalences
in on localizations.
Proof
The left adjoint inclusion is a functor of relative \(\infty \)categories by definition. On the other hand, suppose that is a map in . Via the right localization adjunction, its spatialization fits into a commutative diagram
in , and hence is also in by the twooutofthree property. This shows that the right adjoint is also a functor of relative \(\infty \)categories.
To see that these adjoints induce inverse equivalences on localizations, note that the composite
is the identity, while the composite
admits a natural weak equivalence in to the identity functor (namely, the counit of the adjunction). Hence, the claim follows from Lemma N.1.24. \(\square \)
To conclude this section, we make a pair of general remarks regarding and . We begin by contextualizing these \(\infty \)categories with respect to Lurie’s theory of enriched \(\infty \)categories, which is described in [8, §1].
Remark 2.21
Lurie’s theory of enriched \(\infty \)categories—which provides a satisfactory, compelling, and apparently complete picture (at least when the enriching \(\infty \)category is equipped with the cartesian symmetric monoidal structure)—is premised on the notion of a distributor, the data of which is simply an \(\infty \)category equipped with a full subcategory (see [8, Definition 1.2.1]).^{Footnote 3} Given such a distributor, one can then define \(\infty \)categories and of Segal space objects and of complete Segal space objects with respect to it: these sit as full (in fact, reflective) subcategories
in which

the subcategory consists of those simplicial objects such that

\(Y_{\bullet }\) satisfies the Segal condition and

(see [8, Definition 1.2.7]), while

the subcategory consists of those objects which additionally satisfy a certain completeness condition (see [8, Definition 1.2.10]).
Thus, plays the role of the “enriching \(\infty \)category”, i.e. the \(\infty \)category containing the homobjects in our enriched \(\infty \)category, while its subcategory provides a home for the “object of objects” of the enriched \(\infty \)category. As in the classical case—indeed, the identity distributor simply has and —, one can already meaningfully extract an enriched \(\infty \)category from a Segal space object, but it is only by restricting to the complete ones that one obtains the desired \(\infty \)category of such.
Now, obviously we have
as Segal simplicial spaces are nothing but Segal space objects with respect to the identity distributor on the \(\infty \)category of simplicial spaces. We can clearly also identify the \(\infty \)category of enriched \(\infty \)categories as
the Segal space objects with respect to the distributor (the embedding of spaces as the constant simplicial spaces).^{Footnote 4} \(^,\) ^{Footnote 5} On the other hand, the subcategory of complete Segal space objects can be identified as the pullback
in which the right vertical functor takes an enriched \(\infty \)category to its “levelwise 0th space” object .
We now explain the source of our interest in the \(\infty \)categories and .
Remark 2.22
First and foremost, the reason we are interested in is because this is the natural target of the “prehammock localization” functor
whose construction constitutes the main ingredient of the construction of the hammock localization functor itself (see Sect. 5). On the other hand, we then restrict to the (coreflective) subcategory since this is a convenient full subcategory of on which the levelwise geometric realization functor
(which is a colimit) preserves the Segal condition (which is defined in terms of limits) [recall (the proof of) Lemma 2.13].^{Footnote 6} Indeed, if our “local geometric realization” functor failed to preserve the Segal condition, it would necessarily destroy all “categoryness” inherent in our objects of study. In turn, this would effectively invalidate our right to declare the hammock simplicial spaces
(see Definition 3.17)—which will of course be the homsimplicial spaces in the hammock localization —as “presentations of homspaces” in any reasonable sense.
For these reasons, Segal simplicial spaces are therefore not really our primary interest. However, since for a Segal simplicial space , the counit of the spatialization right localization adjunction is actually fully faithful in the enriched sense, the hammock localization
will then simultaneously

have the hammock simplicial spaces as its homsimplicial spaces, and

have composition maps which both

directly present composition in its geometric realization, and

manifestly encode the notion of “concatenation of zigzags”.

Of course, it would also be possible to restrict further to the (reflective) subcategory
of complete Segal space objects (recall Remark 2.21). However, this is unnecessary for our purposes, since both the prehammock localization functor and the hammock localization functor will land in \(\infty \)categories (namely and , respectively) which admit canonical relative structures via which they present the \(\infty \)category , thus endowing these constructions with external meaning (which are of course compatible with each other in light of Proposition 2.20). Moreover, as the successive inclusions
respectively admit a left adjoint and a right adjoint, this further restriction would in all probability make for a somewhat messier story.
3 Zigzags and hammocks in relative \(\infty \)categories
In studying relative 1categories and their 1categorical localizations, one is naturally led to study zigzags. Given a relative category and a pair of objects , a zigzag from x to y is a diagram of the form
i.e. a sequence of both forwards and backwards morphisms in (in arbitrary (finite) quantities and in any order) such that all backwards morphisms lie in . Under the 1categorical localization , such a diagram is taken to a sequence of morphisms such that all backwards maps are isomorphisms, so that it is in effect just a sequence of composable (forwards) arrows. Taking their composite, we obtain a single morphism in . In fact, one can explicitly construct in such a way that all of its morphisms arise from this procedure.
It is a good deal more subtle to show, but in fact the same is true of relative \(\infty \)categories and their (\(\infty \)categorical) localizations: given a relative \(\infty \)category , it turns out that every morphism in can likewise be presented by a zigzag in itself. (We prove a precise statement of this assertion as Proposition 3.11.)
The representation of a morphism in by a zigzag in is quite clearly overkill: many different zigzags in will present the same morphism in . For example, we can consider a zigzag as being selected by a morphism of relative \(\infty \)categories, where is a zigzag type which is determined by the shape of the zigzag in question; then, precomposition with a suitable morphism of zigzag types will yield a composite which presents a canonically equivalent morphism in . Thus, in order to obtain a closer approximation to , we should take a colimit of the various spaces of zigzags from x to y indexed over the category of zigzag types.
However, this colimit alone will still not generally capture all the redundancy inherent in the representation of morphisms in by zigzags in . Namely, a natural weak equivalence between two zigzags of the same type (which fixes the endpoints) will, upon postcomposing to the localization , yield a homotopy between the morphisms presented by the respective zigzags. Pursuing this observation, we are thus led to consider certain \(\infty \)categories, denoted \({\underline{\mathbf{m }}}(x,y)\) (for varying zigzag types \({\underline{\mathbf{m }}}\)), whose objects are the \({\underline{\mathbf{m }}}\)shaped zigzags from x to y and whose morphisms are the natural weak equivalences (fixing x and y) between them.
Finally, putting these two observations of redundancy together, we see that in order to approximate the homspace , we should be taking a colimit of the various \(\infty \)categories over the category of zigzag types. In fact, rather than taking a colimit of these \(\infty \)categories, we will take a colimit of their corresponding complete Segal spaces (see §N.2), not within the \(\infty \)category of such but rather within the larger ambient \(\infty \)category in which it is definitionally contained; this, finally, will yield the hammock simplicial space , which (as the notation suggests) will be the homsimplicial space in the hammock localization .^{Footnote 7}
This section is organized as follows.

In Sect. 3.1, we lay some groundwork regarding doublypointed relative \(\infty \)categories, which will allow us to efficiently corepresent our \(\infty \)categories of zigzags.

In Sect. 3.2, we use this to define \(\infty \)categories of zigzags in a relative \(\infty \)category.

In Sect. 3.3, we prove a precise articulation of the assertion made above, that all morphisms in the localization are represented by zigzags in .

In Sect. 3.4, we finally define our hammock simplicial spaces and compare them with the hammock simplicial sets of Dwyer–Kan (in the special case of a relative 1category).

In Sect. 3.5, we assemble some technical results regarding zigzags in relative \(\infty \)categories which will be useful later; notably, we prove that for a concatenation \([{\underline{\mathbf{m }}} ; {\underline{\mathbf{m }}}']\) of zigzag types, we can recover the \(\infty \)category \([{\underline{\mathbf{m }}};{\underline{\mathbf{m }}}'] (x,y)\) via the twosided Grothendieck construction (see Definition G.2.3).
3.1 Doublypointed relative \(\infty \)categories
In this subsection, we make a number of auxiliary definitions which will streamline our discussion throughout the remainder of this paper.
Definition 3.1
A doublypointed relative \(\infty \)category is a relative \(\infty \)category equipped with a map . The two inclusions select objects , which we call the source and the target; we will sometimes subscript these to remove ambiguity, e.g. as and . These assemble into the evident \(\infty \)category, which we denote by
Of course, there is a forgetful functor . We will often implicitly consider a relative \(\infty \)category equipped with two chosen objects as a doublypointed relative \(\infty \)category; on the other hand, we may also write to be more explicit. We write for the full subcategory of doublypointed relative categories, i.e. of those doublypointed relative \(\infty \)categories whose underlying \(\infty \)category is a 1category.
Notation 3.2
Recall from Notation N.1.6 that is a cartesian closed symmetric monoidal \(\infty \)category. With respect to this structure, is enriched and tensored over . As for the enrichment, for any , we define the object
of (where we write and to distinguish between the source and target objects); informally, this should be thought of as the relative \(\infty \)category whose objects are the doublypointed relative functors from to , whose morphisms are the doublypointed natural transformations between these (i.e. those natural transformations whose components at \(s_1\) and \(t_1\) are and , resp.), and whose weak equivalences are the doublypointed natural weak equivalences. Then, the tensoring is obtained by taking and to the pushout
in , with its doublepointing given by the natural map from . We will write
to denote this tensoring.
Notation 3.3
In order to simultaneously refer to the situations of unpointed and doublypointed relative \(\infty \)categories, we will use the notation (and similarly for other related notations). When we use this notation, we will mean for the entire statement to be interpreted either in the unpointed context or the doublypointed context.
Notation 3.4
We will write
to denote either the tensoring of Notation 3.2 in the doublypointed case or else simply the cartesian product in the unpointed case.
3.2 Zigzags in relative \(\infty \)categories
In this subsection we introduce the first of the two key concepts of this section, namely the \(\infty \)categories of zigzags in a relative \(\infty \)category between two given objects.
We begin by defining the objects which will corepresent our \(\infty \)categories of zigzags.
Definition 3.5
We define a relative word to be a (possibly empty) word \({\underline{\mathbf{m }}}\) in the symbols \(\mathbf{A}\) (for “any arbitrary arrow”) and . We will write \(\mathbf{A}^{\circ n}\) to denote n consecutive copies of the symbol \(\mathbf{A}\) (for any \(n \ge 0\)), and similarly for . We can extract a doublypointed relative category from a relative word, which for our sanity we will carry out by reading forwards. So for instance, the relative word defines the doublypointed relative category
We denote this object by . Thus, by convention, the empty relative word determines the terminal object (which is the unique relative word determining a doublypointed relative category whose source and target objects are equivalent). Restricting to the orderpreserving maps between relative words (with respect to the evident ordering on their objects, i.e. starting from s and ending at t), we obtain a (nonfull) subcategory of zigzag types.^{Footnote 8} \(^{,}\) ^{Footnote 9} \(^{,}\) ^{Footnote 10} We will occasionally also use this same relative word notation with the symbol , but the resulting doublypointed relative categories will not be objects of .
Remark 3.6
Let be relative words. Then, their concatenation can be characterized as a pushout
in (as well as in ).
Notation 3.7
For any , we will write to denote the number of times that \(\mathbf{A}\) appears in \({\underline{\mathbf{m }}}\), and we will write to denote the number of times that appears in \({\underline{\mathbf{m }}}\).
Remark 3.8
The localization functor
acts on the subcategory of zigzag types as
in effect, it collapses all the copies of and leaves the copies of \([\mathbf{A}]\) untouched.
We now define the first of the two key concepts of this section, an analog of [1, 5.1].
Definition 3.9
Given a relative \(\infty \)category equipped with two chosen objects , and given a relative word , we define the \(\infty \)category of zigzags in from x to y of type \({\underline{\mathbf{m }}}\) to be
If the relative \(\infty \)category is clear from context, we will simply write \({\underline{\mathbf{m }}}(x,y)\).
3.3 Representing maps in by zigzags in
In this subsection, we take a digression to illustrate that our study of zigzags in relative \(\infty \)categories is wellfounded: roughly speaking, we show that any morphism in the localization of a relative \(\infty \)category is represented by a zigzag in the relative \(\infty \)category itself. We will give the precise assertion as Proposition 3.11. In order to state it, however, we first introduce the following terminology.
Definition 3.10
Let and be relative \(\infty \)categories. We will say that a morphism
in represents the morphism
in induced by the localization functor. We will also say that it represents the morphism
in induced from the previous one by the homotopy category functor. In a slight abuse of terminology, we will moreover say that a zigzag
represents the composite
in , where the map is given by \(0 \mapsto 0\) and \(1 \mapsto  {\underline{\mathbf{m }}}_\mathbf{A}\) (i.e. it corepresents the operation of composition), and likewise for the morphism in the homotopy category of the localization selected by either threefold composite in the commutative diagram
in .
Proposition 3.11
Let be a relative \(\infty \)category, and let be a functor selecting a morphism in its localization. Then, for some relative word , there exists a zigzag which represents F.
We will prove Proposition 3.11 in stages of increasing generality. We begin by recalling that any morphism in the 1categorical localization of a relative 1category is represented by a zigzag.
Lemma 3.12
Let be a relative 1category, and let be a functor selecting a morphism in its 1categorical localization. Then, for some relative word , there exists a zigzag which represents F.
Proof
This follows directly from the standard construction of the 1categorical localization of a relative 1category (see e.g. [1, Proposition 3.1]). \(\square \)
Remark 3.13
Lemma 3.12 accounts for the fundamental role that zigzags play in the theory of relative categories and their 1categorical localizations. We can therefore view Proposition 3.11 as asserting that zigzags play an analogous fundamental role in the theory of relative \(\infty \)categories and their (\(\infty \)categorical) localizations.
Remark 3.14
We can view Lemma 3.12 as guaranteeing the existence of a diagram
for some relative word , in which

the upper dotted arrow is a morphism in ,

the lower dotted arrow is its image under the 1categorical localization functor
and

the map is as in Definition 3.10.
With Lemma 3.12 recalled, we now move on to the case of \(\infty \)categorical localizations of relative 1categories.
Lemma 3.15
Let be a relative 1category, and let be a functor selecting a morphism in its localization. Then, for some relative word , there exists a zigzag which represents F.
Proof
Recall from Remark N.1.29 that we have an equivalence . The resulting postcomposition
of F with the projection to the homotopy category selects a morphism in the 1categorical localization . Hence, by Lemma 3.12, we obtain a diagram
for some relative word , in which

the solid horizontal arrows are as in Remark 3.14,

the upper map in induces the dotted map under the functor , so that

the (lower) square in commutes.
That the resulting composite
is equivalent to the functor follows from Lemma 3.16. Thus, in effect, we obtain a diagram
analogous to the one in Remark 3.14 (only with the 1categorical localizations replaced by the \(\infty \)categorical localizations), which proves the claim. \(\square \)
Lemma 3.16
For any \(\infty \)category and any map , the space of lifts
is connected.
Proof
Since the functor creates the subcategory , there is a connected space of lifts of the maximal subgroupoid \(\{ 0 , 1 \} \simeq [1]^\simeq \subset [1]\). Then, in any solid commutative square
there exists a connected space of dotted lifts by definition of the homotopy category. \(\square \)
With Lemma 3.15 in hand, we now proceed to the fully general case of \(\infty \)categorical localizations of relative \(\infty \)categories.
Proof of Proposition 3.11
Observe that the morphism in induces a postcomposition
selecting a morphism in the \(\infty \)categorical localization of the relative 1category . Hence, by Lemma 3.15, we obtain a solid diagram
for some relative word , in which

the lower right diagonal map is an equivalence by Remark N.1.29,

we moreover obtain the upper dotted arrow from Remark 3.6 by induction, and

we define the lower dotted arrow to be its image under localization.
Now, the resulting composite
fits into a commutative diagram
in . In particular, we have obtained a lift
of the composite
which must therefore be equivalent to F itself by Lemma 3.16. Thus, we obtain a diagram
as in the proof of Lemma 3.15, which proves the claim. \(\square \)
Thus, zigzags play an important role not just in the theory of relative 1categories and their 1categorical localizations, but more generally in the theory of relative \(\infty \)categories and their \(\infty \)categorical localizations.
3.4 Hammocks in relative \(\infty \)categories
For a general relative \(\infty \)category , the representation of a morphism in by a zigzag guaranteed by Proposition 3.11 is clearly far from unique. Indeed, any morphism in gives rise to a composite which presents the same morphism in : in other words, the morphisms in corepresent universal equivalence relations between zigzags in relative \(\infty \)categories (with respect to the morphisms that they represent upon localization).
In order to account for this overrepresentation, we are led to the following definition, the second of the two key concepts of this section, an analog of [1, 2.1].
Definition 3.17
Suppose , and suppose . We define the simplicial space of hammocks (or alternatively the hammock simplicial space) in from x to y to be the colimit
We will extend the hammock simplicial space construction further – and in particular, justify its notation – by constructing the hammock localization
of in Sect. 5 (see Remark 5.5).
We now compare our hammock simplicial spaces of Definition 3.17 with Dwyer–Kan’s classical hammock simplicial sets (in relative 1categories).
Remark 3.18
Suppose that is a relative category. Then, by [1, Proposition 5.5], we have an identification
of the classical simplicial set of hammocks defined in [1, 2.1] as an analogous colimit over the 1categorical nerves of the (strict) categories of zigzags in from x to y.^{Footnote 11} However, there are two reasons that this does not coincide with Definition 3.17.

The colimit computing is taken in the subcategory . This inclusion (being a right adjoint) does not generally commute with colimits.

The functors and do not generally agree, but are only related by a natural transformation
in (see Remark N.2.6).
On the other hand, these two constructions do at least participate in a diagram
in , which induces a span
in . We claim that this span lies in the subcategory , i.e. that it becomes an equivalence upon geometric realization; as we have a commutative triangle
in , this will imply that we have a canonical equivalence
in . We view this as a satisfactory state of affairs, since we are only ultimately interested in simplicial sets/spaces of hammocks as presentations of homspaces, anyways.
To see the claim, note first that since is a left adjoint, it commutes with colimits, and so the left leg of the span lies in by the fact that upon postcomposition with the geometric realization functor , the natural transformation
in becomes a natural equivalence
in (again see Remark N.2.6). By Proposition N.2.4, these geometric realizations of colimits in both evaluate to
Now, in order to compute the geometric realization
we begin by observing that the category has an evident Reedy structure, which one can verify has cofibrant constants, so that the dual Reedy structure on has fibrant constants. Moreover, it is not hard to verify that the functor
defines a cofibrant object of . Hence, the colimit
computes the homotopy colimit in , i.e. the colimit of the composite
The claim then follows from the string of equivalences
in (again appealing to Proposition N.2.4).
Remark 3.19
Dwyer–Kan give a pointset definition of the hammock simplicial set in [1, 2.1], and then prove it is isomorphic to the colimit indicated in Remark 3.18. However, working \(\infty \)categorically, it is essentially impossible to make such an ad hoc definition. Thus, we have simply defined our hammock simplicial space as the colimit to which we would like it to be equivalent anyways.
3.5 Functoriality and gluing for zigzags
In this subsection, we prove that \(\infty \)categories of zigzags are suitably functorial for weak equivalences among source and target objects (see Notation 3.23), and we use this to give a formula for an \(\infty \)category of zigzags of type \([{\underline{\mathbf{m }}};{\underline{\mathbf{m }}}']\), the concatenation of two arbitrary relative words (see Lemma 3.24).
Recall from Remark 3.6 that concatenations of relative words compute pushouts in . This allows for inductive arguments, in which at each stage we freely adjoin a new morphism along either its source or its target. For these, we will want to have a certain functoriality property for diagrams of this shape. To describe it, let us first work in the special case of (instead of ). There, if for instance we have an \(\infty \)category with a chosen object and we use this to define a new \(\infty \)category as the pushout
then for any target \(\infty \)category , the evaluation
will be a cartesian fibration by Corollary T.2.4.7.12 (applied to the functor ). The following result is then an analog of this observation for relative \(\infty \)categories; note that there are now two types of “freely adjoined morphisms” we must consider.
Lemma 3.20
Let , choose any , and suppose we are given any .

1.

(a)
If we form the pushout
in , then the composite restriction
is a cocartesian fibration.

(b)
Dually, if we form the pushout
in , then the composite restriction
is a cartesian fibration.

(a)

2.

(a)
If we form the pushout
in , then the composite restriction
is a cocartesian fibration.

(b)
Dually, if we form the pushout
in , then the composite restriction
is a cartesian fibration.

(a)
Proof
We first prove item 1(b). Applying Corollary T.2.4.7.12 to the functor
and noting that (in a way compatible with the evaluation maps), we obtain that the composite restriction
is a cartesian fibration, as desired. The proof of item 1(a) is completely dual.
We now prove item 2(b). For this, consider the diagram
in which all small rectangles are pullbacks and in which we have introduced the ad hoc notation
for the wide subcategory whose morphisms are those natural transformations whose component at lies in . Observing that (in a way compatible with the evaluation maps), it follows from applying Corollary T.2.4.7.12 to the functor
that the composite
is a cartesian fibration, for which the cartesian morphisms are precisely those that are sent to equivalences under the restriction functor
Then, by Propositions T.2.4.2.3(2) and T.2.4.1.3(2), the functor
is also a cartesian fibration, for which any morphism that is sent to an equivalence under the composite
is cartesian. Now, for any map in and any object
choose such a cartesian morphism
Since by definition , it follows that this is in fact a morphism in the (wide) subcategory . Hence, we obtain a diagram
in , in which the right square is a pullback since \(\tilde{\varphi }\) is a cartesian morphism. Moreover, again using the fact that , it is easy to check that the left square is also a pullback. So the entire rectangle is a pullback, and hence \(\tilde{\varphi }\) is also a cartesian morphism for the functor
From here, it follows from the fact that is a subcategory that this functor is indeed a cartesian fibration. The proof of item 2(a) is completely dual. \(\square \)
Given an arbitrary doublypointed relative \(\infty \)category and some relative \(\infty \)category which we consider to be doublypointed via some choice of a pair of objects, we will be interested in the functoriality of the construction
in the variable but for a fixed choice of (or vice versa). This functoriality will be expressed by a variant of Lemma 3.20. However, in order to accommodate the fixing of just one of the two chosen objects, we must first introduce the following notation.
Notation 3.21
Let , let , and let . Then, we write
and
We now give a “halfdoublypointed” variant of Lemma 3.20, but stated only in the special case that we will need.
Lemma 3.22
Let , let , and let

1.
The functor

(a)
is a cocartesian fibration if \({\underline{\mathbf{m }}}\) begins with , and

(b)
is a cartesian fibration if \({\underline{\mathbf{m }}}\) begins with \(\mathbf{A}\).

(a)

2.
The functor

(a)
is a cartesian fibration if \({\underline{\mathbf{m }}}\) ends with , and

(b)
is a cocartesian fibration if \({\underline{\mathbf{m }}}\) ends with \(\mathbf{A}\).

(a)
Proof
If we simply have \({\underline{\mathbf{m }}} = [\mathbf{A}]\) or then these statements follow trivially from Lemma 3.20, so let us assume that the relative word \({\underline{\mathbf{m }}}\) has length greater than 1.
To prove item 2(a), suppose that . Then we have a pullback square
which, making the identification of with in a way which switches the source and target objects, is equivalently a pullback square
From here, the proof parallels that of Lemma 3.20(1)(b), only now we apply Corollary T.2.4.7.12 to the functor
The proof of item 1(a) is completely dual.
To prove item 1(b), let us now suppose that \({\underline{\mathbf{m }}} = [\mathbf{A};{\underline{\mathbf{m }}}']\). Then we have a diagram
in which all small rectangles are pullbacks, almost identical to that of the proof of Lemma 3.20(2)(b). From here, the proof proceeds in a completely analogous way to that one. The proof of item 2(b) is completely dual. \(\square \)
Lemma 3.22, in turn, enables us to make the following definitions.
Notation 3.23
Let , let , and let .

If \({\underline{\mathbf{m }}}\) begins with , we write
for the functor classifying the cocartesian fibration of Lemma 3.22(1)(a). On the other hand, if \({\underline{\mathbf{m }}}\) begins with \(\mathbf{A}\), we write
for the functor classifying the cartesian fibration of Lemma 3.22(1)(b).

If \({\underline{\mathbf{m }}}\) ends with , we write
for the functor classifying the cartesian fibration of Lemma 3.22(2)(a). On the other hand, if \({\underline{\mathbf{m }}}\) ends with \(\mathbf{A}\), we write
for the functor classifying the cocartesian fibration of Lemma 3.22(2)(b).

By convention and for convenience, if is the empty relative word (which defines the terminal relative \(\infty \)category), we let both \({\underline{\mathbf{m }}}(x,)\) and \({\underline{\mathbf{m }}}(,y)\) denote either functor
or
Using Notation 3.23, we now express the \(\infty \)category of zigzags in from x to y of the concatenated zigzag type in terms of the twosided Grothendieck construction (see Definition G.2.3). This is an analog of [1, 9.4].^{Footnote 12}
Lemma 3.24
Let . Then for any and any , we have an equivalence
which is natural in .
Proof
Recall from Remark 3.6 that we have a pushout square
in , through which \([{\underline{\mathbf{m }}};{\underline{\mathbf{m }}}']\) acquires its source object from \({\underline{\mathbf{m }}}\) and its target object from \({\underline{\mathbf{m }}}'\). This gives rise to a string of equivalences
in . From here, the first and second cases follow from Lemma 3.22, Notation 3.23, and Definition G.2.3, while the third and fourth cases follow by additionally appealing to Example G.1.9 and Example G.2.3. \(\square \)
4 Homotopical threearrow calculi in relative \(\infty \)categories
In the previous section, given a relative \(\infty \)category , we introduced the hammock simplicial space
for two objects . The definition of this simplicial space is fairly explicit, but it is nevertheless quite large. In this section, we show that under a certain condition—namely, that admits a homotopical threearrow calculus—we can at least recover this simplicial space up to weak equivalence in (i.e. we can recover its geometric realization) from a much smaller simplicial space, in fact from one of the constituent simplicial spaces in its defining colimit. This condition is often satisfied in practice; for example, it holds when admits the additional structure of a model \(\infty \)category (see Lemma M.8.2).
This section is organized as follows.

In Sect. 4.1, we define what it means for a relative \(\infty \)category to admit a homotopical threearrow calculus, and we state the fundamental theorem of homotopical threearrow calculi (Theorem 4.4) described above.

In Sect. 4.2, in preparation for the proof of Theorem 4.4, we assemble some auxiliary results regarding relative \(\infty \)categories.

In Sect. 4.3, in preparation for the proof of Theorem 4.4, we assemble some auxiliary results regarding ends and coends.
4.1 The fundamental theorem of homotopical threearrow calculi
We begin with the main definition of this section, whose terminology will be justified by Theorem 4.4; it is a straightforward generalization of [5, Definition 4.1], which is itself a minor variant of [1, 6.1(i)].
Definition 4.1
Let . We say that admits a homotopical threearrow calculus if for all and for all \(i,j \ge 1\), the map
in obtained by collapsing the middle weak equivalence induces a map
in (i.e. it becomes an equivalence upon applying the groupoid completion functor ).
Notation 4.2
Since it will appear repeatedly, we make the abbreviation for the relative word
Definition 4.3
For any relative \(\infty \)category and any objects , we will refer to
as the \(\infty \)category of threearrow zigzags in from x to y.
We now state the fundamental theorem of homotopical threearrow calculi, an analog of [1, Proposition 6.2(i)]; we will give its proof in Sect. 4.4.
Theorem 4.4
If admits a homotopical threearrow calculus, then for any , the natural map
in becomes an equivalence under the geometric realization functor .
4.2 Supporting material: relative \(\infty \)categories
In this subsection, we give two results regarding relative \(\infty \)categories which will be used in the proof of Theorem 4.4. Both concern corepresentation, namely the effect of the functor
on certain data in (for a given relative \(\infty \)category ).
Lemma 4.5
Given a pair of maps in , a morphism between them in induces, for any , a natural transformation between the two induced functors
Proof
First of all, the morphism in is selected by a map ; this is equivalent to a map
in , which is adjoint to a map
in . Then, for any , composing with this map yields a functor
which is adjoint to a map
which selects a natural transformation between the two induced functors
as desired. \(\square \)
Lemma 4.6
Let , and form any pushout diagram
in , where the left map is the unique map in . Note that the two possible retractions in of the given map induce retractions in . Then, for any , the induced map
becomes an equivalence under the functor , with inverse given by either map
in induced by one of the given retractions.
Proof
Note that both composites
(of one of the two possible retractions followed by the given map) are connected to by a map in
In turn, both composites
are connected to by a map in . Hence, the result follows from Lemmas 4.5 and N.1.26. \(\square \)
4.3 Supporting material: co/ends
In this subsection, we give a few results regarding ends and coends which will be used in the proof of Theorem 4.4. For a brief review of these universal constructions in the \(\infty \)categorical setting, we refer the reader to [3, §2].
We begin by recalling a formula for the space of natural transformations between two functors.
Lemma 4.7
Given any and any , we have a canonical equivalence
Proof
This appears as [4, Proposition 2.3] (and as [3, Proposition 5.1]). \(\square \)
We now prove a “ninja Yoneda lemma”.^{Footnote 13}
Lemma 4.8
If is an \(\infty \)category equipped with a tensoring , then for any functor , we have an equivalence
in .
Proof
For any test objects and , we have a string of natural equivalences
where the first line follows from the definition of a coend as a colimit (see e.g. [3, Definition 2.5]), the second line uses the tensoring, the third line follows from Lemma 4.7, and the last line follows from the usual Yoneda lemma (Proposition T.5.1.3.1). Hence, again by the Yoneda lemma, we obtain an equivalence
which is natural in . \(\square \)
Then, we have the following result on the preservation of colimits.^{Footnote 14}
Lemma 4.9
If is an \(\infty \)category equipped with a tensoring , then for any functor , the functor
is a left adjoint.
Proof
It suffices to check that for every , the functor
is representable. For this, given any we compute that
where the first line follows from the definition of a co/end as a co/limit (again see e.g. [3, Definition 2.5]), the second line uses the tensoring, and the last line follows from Lemma 4.7. \(\square \)
4.4 The proof of Theorem 4.4
Having laid out the necessary supporting material in the previous two subsection, we now proceed to prove the fundamental theorem of homotopical threearrow calculi (Theorem 4.4). This proof is based closely on that of [1, Proposition 6.2(i)], although we give many more details (recall Remark 1.2).
Proof of Theorem 4.4
We will construct a commutative diagram
in , i.e. a commutative square in which the bottom arrow is equipped with a retraction and in which moreover the top and right map are equivalences. Note that by definition, the object on the bottom left is precisely ; the left map will be the natural map referred to in the statement of the result. The equivalences in satisfy the twooutofsix property, and applying this to the composable sequence of arrows , we deduce that \(\alpha \) is also an equivalence, proving the claim.
We will accomplish this by running through the following sequence of tasks.

1.
Define the two objects on the right.

2.
Define the maps in the diagram.

3.
Explain why the square commutes.

4.
Explain why \(\rho \) gives a retraction of .

5.
Explain why the map \(\beta \) is an equivalence.

6.
Explain why the map is an equivalence.
We now proceed to accomplish these tasks in order.

1.
We define endofunctors by the formulas
and
Then, the object in the upper right is given by
and the object in the bottom right is given by

2.
We define the two evident natural transformations (given by collapsing the two newly added copies of ) and (given by collapsing all internal copies of ) in ; these induce natural transformations and in .^{Footnote 15} We then define the maps in the diagram as follows.

The left map is obtained by taking the geometric realization of the inclusion
into the colimit at the object .

The top map is obtained by taking the geometric realization of the inclusion
into the colimit at the object . (Note that in .)

The right map is obtained by taking the geometric realization of the map
on colimits induced by the natural transformation in .

The bottom map in the square (i.e. the straight bottom map) is obtained by taking the geometric realization of the map
on colimits induced by the natural transformation in .

The curved map is obtained by taking the geometric realization of the map
on colimits induced by the functor


3.
The upper composite in the square is given by the geometric realization of the composite
of the equivalence induced by the component of at the object (which is an isomorphism in ) followed by the inclusion into the colimit at \([\mathbf{A}]\). So, via the (unique) identification \({\underline{\mathbf{3 }}} \cong F([\mathbf{A}])\), we can identify this composite with the inclusion into the colimit at . Meanwhile, the lower composite in the square is given by the geometric realization of the composite
of the map induced by the component of \(\varphi ^{op}\) at \({\underline{\mathbf{3 }}}\) followed by the inclusion into the colimit at \({\underline{\mathbf{3 }}}\). Now, the map in is given by
On the other hand, applying F to the unique map in , we obtain a map in given by
which corepresents a map
in which participates in the diagram
defining . So, in order to witness the commutativity of the square, it suffices to obtain an equivalence between the two maps
But there is an evident cospan in between the two maps \(\varphi _{\underline{\mathbf{3 }}}\) and \(F(\gamma )\), so this follows from Lemma 4.5, Lemma N.1.26, and Proposition N.2.4.

4.
The fact that follows from applying Proposition G.2.5 to the diagram
and invoking Proposition N.2.4 to obtain a retraction diagram

5.
We first claim that for any , the map
is an isomorphism. Indeed, note that by Proposition G.2.1, we have an equivalence
The category
admits a span of natural transformations from the identity functor to its fiber over the object , whose component at an object is indicated by the natural commutative diagram
in (in which the dotted arrow is simply the extension of the upper map over an isomorphism).^{Footnote 16} Hence, by Lemma N.1.26 the inclusion of the fiber over induces an equivalence upon groupoid completions. But this fiber is precisely .
Now, assembling the above observation over all , we see that the map
is an equivalence in . Using this, and denoting by the evident tensoring
we obtain the map
as string of equivalences
in , in which

the second and fifth lines are purely for notational convenience,

we apply to the functor

Lemma 4.8 to obtain the first line,

Lemma 4.9 to obtain the fourth line, and

Lemma 4.8 again to obtain the last line,
and


the third line follows from the equivalence in obtained above.
(So in fact, the map \(\beta \) itself is already an equivalence in (i.e. before geometric realization).)

6.
We claim that for every the map
in becomes an equivalence after geometric realization. This follows from an analysis of the corepresenting map in : it can be obtained as a composite
in , in which each \({\underline{\mathbf{m }}}'_i\) is obtained from \({\underline{\mathbf{m }}}'_{i1}\) by omitting one of the internal appearances of in \(F({\underline{\mathbf{m }}})\), and the corresponding map is obtained by collapsing this copy of to an identity map. Each map
in becomes an equivalence after geometric realization, by Lemma 4.6 when the abouttobeomitted appearance of in \({\underline{\mathbf{m }}}'_{i1}\) is adjacent to another appearance of , and by applying the definition of admitting a homotopical threearrow calculus (Definition 4.1) to (either one or two iterations, depending on the shape of \({\underline{\mathbf{m }}}'_{i1}\), of) the combination of Lemma 3.24 and Proposition G.2.4. Hence, the composite map
which is precisely the map , does indeed become an equivalence upon geometric realization as well. Then, since colimits commute, it follows that the induced map
is an equivalence in . \(\square \)
5 Hammock localizations of relative \(\infty \)categories
In Sect. 3, given a relative \(\infty \)category and a pair of objects , we defined the corresponding hammock simplicial space
(see Definition 3.17). In this section, we proceed to globalize this construction, assembling the various hammock simplicial spaces of into a Segal simplicial space—and thence a enriched \(\infty \)category—whose compositions encode the concatenation of zigzags in .
The bulk of the construction of the hammock localization consists in constructing the prehammock localization: this will be a Segal simplicial space
whose nth level is given by the colimit
For clarity, we proceed in stages.
First, we build an object which simultaneously corepresents

all possible sequences (of any length) of composable zigzags, and

all possible concatenations among these sequences.
Construction 5.1
Observe that is a monoid object, i.e. a monoidal category: its multiplication is given by the concatenation functor
and the unit map selects the terminal object .^{Footnote 17} We can thus define its bar construction
which has (so that ), with face maps given by concatenation and with degeneracy maps given by the unit. This admits an oplax natural transformation to the functor
which we encode as a commutative triangle
in (recall Definition G.3.1 and Example G.1.15): in simplicial degree n, this is given by the iterated concatenation functor
(which in degree 0 is simply the composite
i.e. the inclusion of the terminal object ).^{Footnote 18} \(^,\) ^{Footnote 19} Taking opposites, we obtain a commutative triangle
in , which now encodes a lax natural transformation from the bar construction
on the monoid object (note that the involution is covariant) to the functor
We now map into an arbitrary relative \(\infty \)category and extract the indicated colimits, all in a functorial way.
Construction 5.2
A relative \(\infty \)category represents a composite functor
Considering this as a natural transformation in , we can postcompose it with the lax natural transformation obtained in Construction 5.1, yielding a composite lax natural transformation encoded by the diagram
in . Then, by Proposition T.4.2.2.7, there is a unique “fiberwise colimit” lift in the diagram
in .^{Footnote 20} Thus, the resulting composite
takes each object to the colimit of the composite
We denote this simplicial object in simplicial spaces by
Allowing to vary, this assembles into a functor
We now show that the bisimplicial spaces of Construction 5.2 are in fact Segal simplicial spaces.
Lemma 5.3
For any , the object satisfies the Segal condition.
Proof
We must show that for every \(n \ge 2\), the nth Segal map
(to the nfold fiber product) is an equivalence in . As is an \(\infty \)topos, colimits therein are universal, i.e. they commute with pullbacks [see Definition T.6.1.0.4 and Theorem T.6.1.0.6 (and the discussion at the beginning of §T.6.1.1)]. Moreover, note that we have a canonical equivalence in . Hence, by induction, we have a string of equivalences
(where in the penultimate line we appeal to Fubini’s theorem for colimits) which, chasing through the definitions, visibly coincides with the nth Segal map. This proves the claim. \(\square \)
We finally come to the main point of this section.
Definition 5.4
By Lemma 5.3, the functor given in Construction 5.2 admits a factorization
through the \(\infty \)category of Segal simplicial spaces. We again denote this factorization by
and refer to it as the prehammock localization functor.^{Footnote 21} Then, we define the hammock localization functor
to be the composite
Remark 5.5
Given a relative \(\infty \)category , the 0th level of its prehammock localization
is given by
which is simply the nerve of the subcategory of weak equivalences. Thus, its space of objects is simply
Moreover, unwinding the definitions, it is manifestly clear that

its homsimplicial spaces are precisely the hammock simplicial spaces of (recall Definitions 2.8 and 3.17), and

its compositions correspond to concatenation of zigzags (with identity morphisms corresponding to zigzags of type ).
Of course, we have a canonical counit weak equivalence
in which is even fully faithful in the enriched sense, so that the hammock localization enjoys all these same properties.
Just as in the 1categorical case, the hammock localization of admits a natural map from .
Construction 5.6
Returning to Construction 5.1, observe that there is a tautological section
which takes to , and which takes a map in to the map corresponding to the fiber map which, in the ith factor of , is given by the unique map
in . This is opposite to a tautological section
which gives rise to a composite map
admitting a natural transformation to the standard inclusion (as the “target” factor, i.e. the fiber over \(1 \in [1]\)). This postcomposes with the composite
appearing in Construction 5.2 to give a natural transformation
in .^{Footnote 22} Thus, in simplicial degree n, this map is simply the inclusion into the colimit defining at the object
Restricting levelwise to (the nerve of) the maximal subgroupoid, we obtain a composite
As this source lies in , we obtain a canonical factorization
in . This clearly assembles into a natural transformation
in .
Definition 5.7
For a relative \(\infty \)category , we refer to the map
in of Construction 5.6 as its tautological inclusion.
We end this section with the following fundamental result, an analog of [1, Proposition 3.3]. In essence, it shows that when considered as morphisms in the hammock localization, weak equivalences in both represent and corepresent equivalences in the underlying \(\infty \)category. Just as with the fundamental theorem of homotopical threearrow calculi (Theorem 4.4), its proof will be substantially more involved than that of its 1categorical analog (recall Remark 1.2).
Proposition 5.8
Let , and let . Suppose we are given a weak equivalence
and let us also denote by the resulting composite morphism
Then, the induced “composition with w” maps
and
in become equivalences in upon geometric realization. Moreover, if we denote by the composite morphism
then their inverses are respectively given by the geometric realizations of the induced “composition with \(w^{1}\)” maps
and
in .
Proof
We prove the first statement; the second statement follows by a nearly identical argument. Moreover, we will only show that the composite map
is an equivalence; that the composite
is an equivalence will follow from a very similar argument.
For each , let us define a functor
given informally by taking a zigzag
in to the zigzag
in , in which both new maps are the chosen weak equivalence w.^{Footnote 23} This operation is clearly natural in , i.e. it assembles into a natural transformation
Then, using Proposition N.2.4 and the fact that the geometric realization functor commutes with colimits (being a left adjoint), we see that the composite
is obtained as the composite
To see that this is an equivalence, for each let us define a map in to be opposite the map in which collapses the newly concatenated copy of to the map . These assemble into a natural transformation in , and hence we obtain a natural transformation
Moreover, For each we have a functor
adjoint to a functor
given informally by taking a zigzag
in to the diagram
in representing a morphism in , where the maps in the right two squares are all either the chosen weak equivalence or are . These assemble into a morphism
in , i.e. a modification from to \(\varphi \). By Proposition G.2.8, this induces a natural transformation
which, by Lemma N.1.26 and Proposition G.2.1, gives a homotopy between the maps
and
in . Hence, to show that the above composite is an equivalence, it suffices to show that the composite
is an equivalence. But this composite fits into a commutative triangle
obtained by applying Proposition G.2.5 to the diagram
so it is an equivalence. This proves the claim. \(\square \)
6 From fractions to complete Segal spaces, redux
As an application of the theory developed in this paper, we now provide a sufficient condition for the Rezk nerve of a relative \(\infty \)category to be either

a Segal space or

a complete Segal space,
thus giving a partial answer to our own Question N.3.6, which we refer to as the calculus theorem.^{Footnote 24} This result is itself a direct generalization of joint work with Low regarding relative 1categories (see [5, Theorem 4.11]). That result, in turn, generalizes work of Rezk, Bergner, and Barwick–Kan; we refer the reader to [5, §1] for a more thorough history.
Theorem 6.1
Suppose that admits a homotopical threearrow calculus.

1.
is a Segal space.

2.
Suppose moreover that satisfies the twooutofthree property. Then is a complete Segal space if and only if is saturated.
The proof of the calculus theorem (Theorem 6.1) is very closely patterned on the proof of [5, Theorem 4.11] (the main theorem of that paper), which is almost completely analogous but holds only for relative 1categories.^{Footnote 25} We encourage any reader who would like to understand it to first read that paper: there are no truly new ideas here, only generalizations from 1categories to \(\infty \)categories.
Proof of Theorem 6.1
For this proof, we give a detailed stepbystep explanation of what must be changed in the paper [5] to generalize its main theorem from relative 1categories to relative \(\infty \)categories.

For [5, Definition 2.1], we replace the notion of a “weak homotopy equivalence” of categories by the notion of a map in which becomes an equivalence under (i.e. a Thomason weak equivalence (see Definition G.A.2 and Remark G.A.3)).

The proof of [5, Lemma 2.2] carries over easily using Lemma N.1.26.

For [5, Definition 2.3], we replace the notion of a “homotopy pullback diagram” of categories by the notion of a commutative square in which becomes a pullback square under (i.e. a homotopy pullback diagram in ).

For [5, Definition 2.4], we replace the notions of “Grothendieck fibrations” and “Grothendieck opfibrations” of categories by those of cartesian fibrations and cocartesian fibrations of \(\infty \)categories (see §G.1 and [15]).

For [5, Remark 2.5], as the entire theory of \(\infty \)categories is in essence already only pseudofunctorial, there is no corresponding notion of a co/cartesian fibration being “split” (or rather, every co/cartesian fibration should be thought of as being “split”).

The evident generalization of [5, Example 2.6] can be obtained by applying Corollary T.2.4.7.12 to an identity functor of \(\infty \)categories.

The evident generalization of (the first of the two dual statements of) [5, Theorem 2.7] is proved as Corollary G.4.3.

The evident generalization of [5, Corollary 2.8] again follows directly (or can alternatively be obtained by combining Example N.1.12 and Lemma N.1.20).

For [5, Definition 2.9], we use the definition of the “twosided Grothendieck construction” given in Definition G.2.3. (Note that the 1categorical version is simply the corresponding (strict) fiber product.)

The evident analog of [5, Lemma 2.11] is proved as Proposition G.2.4.

For [5, Definition 3.1], we replace the notion of a “relative category” by the notion of a “relative \(\infty \)category” given in Definition N.1.1; recall from Remark N.1.2 that here we are actually working with a slightly weaker definition. We replace the notion of its “homotopy category” by that of its localization given in Definition N.1.8. We have already defined the notion of a relative \(\infty \)category being “saturated” in Definition N.1.14.

For [5, Definition 3.2], we have already made the analogous definitions in Notation N.1.6.

For [5, Definitions 3.3 and 3.6], we have already made the analogous definitions in Definitions 3.5 and 3.9.

The evident analog of [5, Remark 3.7] is now true by definition (recall Notation 3.2).

For [5, Proposition 3.8], the paper actually only uses part (ii), whose evident analog is provided by Lemma 3.20(1).

For [5, Lemma 3.10], note that the functors in the statement of the result as well as in its proof are all corepresented by maps in ; the proof of the analogous result thus carries over by Lemma 4.5.

For [5, Lemma 3.11], again everything in the statement of the result as well as in its proof are all corepresented; again the proof carries over by Lemma 4.5.

For [5, Definition 4.1], we have already defined a “homotopical threearrow calculus” for a relative \(\infty \)category in Definition 4.1.

For [5, Theorem 4.5], we use the more general but slightly different definition of hammocks given in Definition 3.17 (recall Remark 3.18); part (i) is proved as Theorem 4.4, while part (ii) follows immediately from the definitions, particularly Definitions 5.4 and 2.8. (Note that in the present framework, the “reduction map” is simply replaced by the canonical map to the colimit defining the simplicial space of hammocks.)

For [5, Corollary 4.7], the evident analog of [1, Proposition 3.3] is proved as Proposition 5.8.

For [5, Proposition 4.8], the proof carries over essentially without change. (The functor considered there when proving that the rectangle (AC) is a homotopy pullback diagram is replaced by our functor of Notation 3.23.)

For [5, Lemma 4.9], the map itself in the statement of the result comes from the functoriality
and
of Notation 3.23, as do the vertical maps in the commutative square in the proof. The horizontal maps in that square are corepresented by maps in , and it clearly commutes by construction. The evident analog of [1, Proposition 9.4] is proved as Lemma 3.24.

For [5, Proposition 4.10], note that all morphisms in both the statement of the result and its proof are corepresented by maps in ; the proof itself carries over without change.

For [5, Theorem 4.11] (whose analog is Theorem 6.1 itself), note that we are now proving an \(\infty \)categorical statement (instead of a modelcategorical one), and so there are no issues with fibrant replacement.

The proof of part (1) of Theorem 6.1 is identical to the proof of part (i) there: it follows from our analog of [5, Proposition 4.10].

We address the two halves of the proof of part (2) of Theorem 6.1 in turn.
 \(*\) :

The proof of the “only if” direction runs analogously to that of [5, Theorem 4.11(ii)], only now we use that given two objects in an \(\infty \)category , any path between their postcompositions can be represented by a zigzag connecting them (for some sufficiently large i).
 \(*\) :

We must modify the proof of the “if” direction slightly, as follows. Assume that is saturated. By the local universal property of the Rezk nerve (Theorem N.3.8), we have an equivalence in . Note also that by the twooutofthree assumption, any two objects which select the same path component under the composite
are either both weak equivalences or both not weak equivalences. Now, for any object of , recalling Remark 2.3 and invoking the saturation assumption, we see that the corresponding map selects an equivalence under the postcomposition if and only if it factors as . From here, the proof proceeds identically. \(\square \)

Remark 6.2
After establishing the necessary facts concerning model \(\infty \)categories, we obtain an analog of [5, Corollary 4.12] as Theorem M.10.1.
Remark 6.3
In light of Remark N.3.2, [5, Remark 4.13] is strictly generalized by the local universal property of the Rezk nerve (Theorem N.3.8).
Notes
The word “spatialization” is meant to indicate that the 0th object of its output will lie in the subcategory of constant simplicial spaces.
To see that the inclusion of the full subcategory of constant objects is a distributor, note that if is an \(\infty \)topos and is a full subcategory which is stable under limits and colimits, then is automatically a distributor. The only remaining point is to verify condition (4) of [8, Definition 1.2.1]. The functor is given on objects by , with functoriality given by pullback in . This clearly factors as the composite , in which the latter functor is similarly given by , which then preserves colimits by Proposition T.6.1.3.10 and Theorem T.6.1.3.9.
In contrast with Remark 2.7, enriched \(\infty \)categories do not quite have an analog in ordinary category theory, only in enriched category theory. (It is only a coincidence of the special case presently under study that the two \(\infty \)categories and participating in the distributor appear to be so closely related.)
As the functor is left adjoint to the inclusion and hence in particular commutes with colimits, its application to the hammock simplicial space will yield the aforementioned colimit of \(\infty \)categories. Moreover, since we are ultimately interested in hammock simplicial spaces for their geometric realizations, in view of Proposition N.2.4 we can consider this shift in ambient \(\infty \)category merely as a technical convenience. For instance, there is an evident explicit description of the constituent spaces in the hammock simplicial space [analogous to the 1categorical case (see [1, 2.1])].
Note that the objects of can in fact be considered as strict doublypointed relative categories, and moreover itself can be considered as a strict category. However, as we will only use these objects in invariant manipulations, we will not need these observations.
Omitting the terminal relative word from (and considering it as a strict category), we obtain the opposite of the indexing category of [1, 4.1]. We prefer to include this terminal object: it is the unit object for a monoidal structure on given by concatenation, which will play a key role in the definition of the hammock localization (see Construction 5.1).
Note that an orderpreserving map must lay each morphism \([\mathbf{A}]\) across some \([\mathbf{A}^{\circ m}]\) (for some \(m \ge 0\)), and must lay each morphism across some (for some \(n \ge 0\)). In particular, it cannot lay a morphism \([\mathbf{A}]\) across a morphism (or vice versa, of course).
It is not hard to see that the presence of the initial object (which is what distinguishes this indexing category from ) does not change this colimit.
In the statement of [1, 9.4], the third appearance of \({\underline{\mathbf{m }}}\) should actually be \({\underline{\mathbf{m }}}'\).
The name is apparently due to Leinster (see [6, Remark 2.2]).
Recall that the involution is contravariant on 2morphisms.
Each path component of this category contains exactly one object lying over .
In fact, we can even consider as a monoid object in (i.e. a strict monoidal category), but this is unnecessary for our purposes.
The reason that we must compose with the forgetful functor is that the oplax structure maps (e.g. the inclusion ) do not respect the doublepointings.
It is also true that for a monoidal (\(\infty \))category whose unit object is terminal, the bar construction admits a canonical lax natural transformation to , whose components are again given by the iterated monoidal product. But this is distinct from what we seek here.
The object in the bottom left of this diagram is a “relative join” (see Definition T.4.2.2.1), which in this case actually simply reduces to a “directed mapping cylinder” (see Example G.1.8).
The terminology “prehammock localization” should be parsed as “pre(hammock localization)”: it already contains the hammock simplicial spaces (see Remark 5.5), it is just not itself the hammock localization.
Note that this source is just the image of the Rezk prenerve under the inclusion (recall Definition N.3.1).
This (and subsequent constructions) can easily be made precise by defining a suitable notion of a map in a relative word being forced to land at w; we will leave such a precise construction to the interested reader.
The Rezk nerve is a straightforward generalization of Rezk’s “classification diagram” construction, which we introduced and studied in §N.3.
The 1categorical Rezk nerve and the Rezk nerve of a relative \(\infty \)category are essentially equivalent (see Remark N.3.2), which is why essentially the same proof can be applied in both cases.
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Acknowledgements
We would like to thank David Ayala, Marc Hoyois, Tyler Lawson, Zhen Lin Low, Adeel Khan Yusufzai, and an anonymous referee for their helpful input. We also gratefully acknowledge the financial support from the NSF graduate research fellowship program (Grant DGE1106400) provided during the time that this paper was written.
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Communicated by Mark Behrens.
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MazelGee, A. Hammocks and fractions in relative \(\infty \)categories. J. Homotopy Relat. Struct. 13, 321–383 (2018). https://doi.org/10.1007/s4006201701840
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DOI: https://doi.org/10.1007/s4006201701840