Mapping spaces in quasi-categories

One problem with using the Dwyer-Kan models in our setting is that given $0$-simplices $a$, $b$, and $c$ in a quasi-category $K$, there do not exist naturally arising composition maps

In [L] Lurie gives a functor $\mathfrak{C}\colon{s{\mathcal{S}et}}\rightarrow s{\mathcal{C}at}$, and the simplicial sets $\mathfrak{C}(K)(a,b)$ can be thought of as models for function complexes of $K$ which do admit such composition maps. In [DS1] we gave another functor $\mathfrak{C}^{nec}$ which accomplished the same task, and we proved that the mapping spaces $\mathfrak{C}(K)(a,b)$ and $\mathfrak{C}^{nec}(K)(a,b)$ were naturally Kan equivalent. In fact we gave an entire collection of different models $\mathfrak{C}^{{\mathcal{G}}}$, all of which were Kan equivalent. What is not immediately clear is how to connect the Dwyer-Kan function complexes to the mapping spaces arising in $\mathfrak{C}$, $\mathfrak{C}^{nec}$, or $\mathfrak{C}^{{\mathcal{G}}}$. This is explained in Section 5, where they are shown to be connected by a canonical zig-zag of Kan equivalences.

At one level the connection can be seen as follows. Recall from [DS1] that a necklace is a simplicial set of the form $\Delta^{n_{1}}\vee\cdots\vee\Delta^{n_{k}}$, obtained from simplices by gluing the final vertex of one to the initial vertex of its successor. There is a natural linear order on the vertices of a necklace, and there is a unique 1-simplex connecting each non-terminal vertex to its successor. The spine $\Spi[T]$ of a necklace $T$ is the union of these 1-simplices, and the associated simplex $\Delta[T]$ of $T$ is the simplex with the same ordered vertex set as $T$.

The mapping space $\mathfrak{C}^{nec}(K)(a,b)$ is just the nerve of the evident category whose objects are pairs $[T,T\rightarrow K_{a,b}]$ where $T$ is a necklace and $T\rightarrow K$ is a map sending the initial and final vertices of $T$ to $a$ and $b$. The mapping spaces $\mathfrak{C}^{{\mathcal{G}}}(a,b)$ are defined similarly, but where one replaces the category of necklaces with a category ${\mathcal{G}}$ of some other suitable “gadgets.”

For any necklace $T$, there is a canonical inclusion $T\hookrightarrow\Delta[T]$; it may be checked that this inclusion is a weak equivalence in ${s{\mathcal{S}et}}_{J}$ (see Lemma 9.2). Any map $T\rightarrow K$ therefore gives rise to a zig-zag

where the map $\Delta^{1}\rightarrow\Delta[T]$ is the unique $1$-simplex connecting the initial and final vertices of the simplex $\Delta[T]$. Zig-zags of the above type are known to give a model for homotopy function complexes (see Section 3).

The considerations from the above paragraph lead to a comparison map between the $\mathfrak{C}(K)(a,b)$ mapping spaces and the Dwyer-Kan mapping spaces. This map turns out to be a Kan equivalence, although it should be noted that our proof of this is not direct: it is a crafty argument using one of the $\mathfrak{C}^{{\mathcal{G}}}$ constructions where one replaces necklaces by a more general class of gadgets. See Section 5 for the details. In any case, we obtain the following (where one may choose any standard model for the homotopy function complex):

The previous theorem is a key step in the proof of the following important result:

We prove the above theorem as Corollary 8.2.

Both Theorem 1.4 and 1.5 are originally due to Lurie, and are proven in [L]. In Lurie’s book he starts by developing the properties of mapping spaces and rigidification and then proves the existence of the Joyal model structure as a consequence of this work. His methods involve a detailed and lengthy study of what he calls “straightening and unstraightening” functors, and it was a vague dissatisfaction with this material—together with the hope of avoiding it—that first led us to the work in the present paper.

In the present paper we start with the Joyal model structure. In both [L] and [J2] it takes over a hundred pages to prove its existence, but by boiling things down to the bare essentials (and following the method of Joyal) we are able to give a streamlined approach and create the structure fairly quickly. At that point we immediately have the Dwyer-Kan techniques at our disposal, and it is through the use of these that we develop the properties of mapping spaces and rigidification. Thus, in some ways our approach is opposite that of [L].

Due to the inherent differences in the two approaches, it is slightly awkward for us to quote results from [L] without creating confusions and possible circularities. Because of this, there are a few minor results whose proofs we end up repeating or redoing in a slightly different way. The result is that the present paper can be read independently of [L]—although this should not be taken as a denial of the intellectual debt we owe to that work.

The material on quasi-categories needed in the paper is all reviewed in Section 2. This material is mostly due to Joyal [J2], and can also be found in Lurie [L]. Despite this, we have included three appendices in the paper where we prove all the assertions from Section 2. The reason for this is that although the results we need are reasonably straightforward, in both [L] and [J2] they are intertwined with so many other things that it is difficult for a reader to extract their proofs. Especially in the case of the Joyal model structure, a basic tool in our entire approach, we felt that it was important to have a relatively short and self-contained argument. The fourteen pages in our appendices are a bit longer than ideal, but quite a bit shorter than the currently available alternatives.

Let us outline the rest of the paper. In Section 3, we review homotopy function complexes in general model categories ${\mathcal{M}}$. We apply these ideas in Section 4 to the case ${\mathcal{M}}=(\partial\Delta^{1}\downarrow{s{\mathcal{S}et}}_{J})$. That is, the function complex of maps $\Delta^{1}\rightarrow X$ in ${\mathcal{M}}$ gives the “internal mapping space” between vertices $a$ and $b$ in a quasi-category $X$. In Section 5 we compare these internal mapping spaces to the rigidification mapping spaces $\mathfrak{C}(X)(a,b)$, and in so doing rely heavily on [DS1]. In Section 6 we define relative mapping spaces (the map $\partial\Delta^{1}\rightarrow\Delta^{1}$ is the thing being generalized here). In Section 7 we introduce the notion of DK-equivalence in ${s{\mathcal{S}et}}$, which is somehow intermediate between $\mathfrak{C}$-equivalence and Joyal equivalence. In Section 8 we prove that these three notions of equivalence agree. Section 8 also provides a new proof of Lurie’s result that there is Quillen equivalence between ${s{\mathcal{S}et}}_{J}$ and ${s{\mathcal{C}at}}$ (see Corollary 8.2). Finally, in Section 9 we prove some technical results that are stated without proof in Sections 1 and 4.

We will use ${s{\mathcal{S}et}}_{K}$ to refer to the usual model structure on simplicial sets, which we’ll term the Kan model structure. The fibrations are the Kan fibrations, the weak equivalences (called Kan equivalences from now on) are the maps which induce homotopy equivalences on geometric realizations, and the cofibrations are the monomorphisms.

Quasi-categories are simplicial sets, and the mapping spaces between their vertices are also simplicial sets. Quasi-categories will always be viewed within the Joyal model structure, and their mapping spaces will always be viewed within the Kan model structure.

We will often be working with the category ${s{\mathcal{S}et}}_{*,*}=(\partial\Delta^{1}\downarrow{s{\mathcal{S}et}})$. When we consider it as a model category, the model structure we use will always be the one imported from the Joyal model structure on ${s{\mathcal{S}et}}$; we will denote it $({s{\mathcal{S}et}}_{*,*})_{J}=(\partial\Delta^{1}\downarrow{s{\mathcal{S}et}}_{J})$, or just ${s{\mathcal{S}et}}_{*,*}$. When we write $\Delta^{1}$ as an object of ${s{\mathcal{S}et}}_{*,*}$, we always mean to use the canonical inclusion $\partial\Delta^{1}\rightarrow\Delta^{1}$.

An object of ${s{\mathcal{S}et}}_{*,*}$ is a simplicial set $X$ with two distinguished points $a$ and $b$. We sometimes (but not always) write $X_{a,b}$ for $X$, to remind ourselves that things are taking place in ${s{\mathcal{S}et}}_{*,*}$ instead of ${s{\mathcal{S}et}}$.

When ${\mathcal{C}}$ is a category we write ${\mathcal{C}}(a,b)$ instead of $\Hom_{{\mathcal{C}}}(a,b)$. We reserve the $\Hom_{K}(a,b)$ notation for when $K$ is a quasi-category, to denote various models of the homotopy function complex. See Proposition 1.2 or Section 4.3.

Finally, in certain places we have to deal with simplices and horns. The horn $\Lambda^{n}_{k}$ is the union of all faces containing the vertex $k$. If $\{m_{0},\ldots,m_{k}\}$ is an ordered set, we use $\Delta^{\{m_{0},\ldots,m_{k}\}}$ as notation for the $k$-simplex whose vertices are $m_{0},\ldots,m_{k}$. Likewise, the “$m_{i}$-face” of $\Delta^{\{m_{0},\ldots,m_{k}\}}$ is the codimension one face not containing the vertex $m_{i}$, and $\Lambda^{\{m_{0},\ldots,m_{k}\}}_{m_{i}}$ denotes the horn consisting of all faces that contain $m_{i}$. If $j_{0}\leq\cdots\leq j_{n}$, we may use “bracket notation” and write $[m_{j_{0}},\ldots,m_{j_{n}}]$ for the subsimplex of $\Delta^{\{m_{0},\ldots,m_{k}\}}$ having vertices $m_{j_{0}},\ldots,m_{j_{n}}$.

For a quasi-category $X$, we refer to the $0$-simplices $a\in X_{0}$ as objects; to the $1$-simplices $f\in X_{1}$ as morphisms, which we denote by $f\colon a\rightarrow b$ if $d_{1}(f)=a,d_{0}(f)=b$; and to the degenerate 1-simplices $s_{0}(a)$ as identity morphisms, which we denote by $\textnormal{id}_{a}$.

For any set $S$, let $ES$ be the $0$-coskeleton of the set $S$. Write $E^{n}$ for $E(\{0,1,\ldots,n\})$. Note that $E^{1}$ has exactly two non-degenerate simplices in each dimension, and that its geometric realization is the usual model for $S^{\infty}$.

Let $X$ be a quasi-category. An $\infty$-isomorphism in $X$ is a map $E^{1}\rightarrow X$. We sometimes abuse terminology and say that a $1$-simplex $\Delta^{1}\rightarrow X$ is an $\infty$-isomorphism if it extends to a map $E^{1}\rightarrow X$.

There is a closely related concept called quasi-isomorphism; a quasi-isomorphism is a map $\sk_{2}(E^{1})\rightarrow X$. The simplicial set $\sk_{2}(E^{1})$ consists of two nondegenerate $1$-simplices and two nondegenerate $2$-simplices. In other words, a quasi-isomorphism in $X$ consists of two objects $a$ and $b$, maps $f\colon a\rightarrow b$ and $g\colon b\rightarrow a$, and two $2$-simplices of the form

Again, we sometimes abuse terminology and say that a $1$-simplex in $X$ is a quasi-isomorphism if it extends to a map $\sk_{2}(E^{1})\rightarrow X$.

The following results record the key properties we will need concerning $\infty$-isomorphisms and quasi-isomorphisms:

Note that part (b) implies that one may compose (in a non-unique way) $\infty$-isomorphisms to yield another $\infty$-isomorphism.

The proofs of Proposition 2.2 and 2.3(a) are not at all straightforward. They hinge on a lemma discovered by Joyal which we also record here (although the proof is again deferred to Appendix B).

It is easy to see that in a Kan complex every morphism is a quasi-isomorphism (use Proposition 2.2). Conversely, if in a certain quasi-category $X$ every morphism is a quasi-isomorphism then Proposition 2.4 implies that $X$ is a Kan complex. Thus one has the slogan “Kan complexes are $\infty$-groupoids.”

For a quasi-category $X$, let $J(X)\subseteq X$ be the subcomplex consisting of those simplices having the property that every 1-dimensional face is a quasi-isomorphism. If ${\mathcal{K}an}\subseteq{s{\mathcal{S}et}}$ denotes the full subcategory of Kan complexes, then $J\colon{s{\mathcal{S}et}}\rightarrow{\mathcal{K}an}$ and this is the right adjoint of the inclusion functor ${\mathcal{K}an}\hookrightarrow{s{\mathcal{S}et}}$. In particular, note that $J$ preserves limits. The quasi-category $J(X)$ is the “$\infty$-groupoid of $\infty$-isomorphisms in $X$.”

A map of simplicial sets will be called inner anodyne if it can be constructed out of the maps $\Lambda^{n}_{k}\hookrightarrow\Delta^{n}$, $0<k<n$, by cobase changes and compositions. Clearly any quasi-category has the right-lifting-property with respect to all inner anodyne maps.

Given morphisms $f\colon A\rightarrow B$ and $g\colon C\rightarrow D$, let $f\square g$ denote the induced map

A map $X\rightarrow Y$ is an inner fibration if it has the right-lifting-property with respect to the inner horn inclusions. The following result is an immediate consequence of Proposition 2.6, using adjointness:

Let $A$ and $Z$ be simplicial sets, and assume $Z$ is a quasi-category. Define a relation on ${s{\mathcal{S}et}}(A,Z)$ by saying that $f\sim g$ if there exists a map $H\colon A\times E^{1}\rightarrow Z$ such that $Hi_{0}=f$ and $Hi_{1}=g$, where $i_{0},i_{1}\colon A\hookrightarrow A\times E^{1}$ are the evident inclusions. It is easy to see that this relation is both reflexive and symmetric, and Proposition 2.3(b) implies that it is also transitive (using that $Z^{A}$ is a quasi-category). We write $[A,Z]_{E^{1}}$ to denote the set ${s{\mathcal{S}et}}(A,Z)/\sim$.

Like all the results in this section, the following is due to Joyal. It is proven in Appendix C.

The model category structure provided by Theorem 2.13 will be denoted ${s{\mathcal{S}et}}_{J}$. Note that it is left proper because every object is cofibrant. We will use the term Joyal fibration for the fibrations in ${s{\mathcal{S}et}}_{J}$.

The following result will be used frequently:

We recall from [L] that there are adjoint functors

Here $\mathfrak{C}(\Delta^{n})$ is a specific simplicial category one writes down, and $N({\mathcal{D}})$ is the simplicial set $[n]\mapsto{s{\mathcal{C}at}}(\mathfrak{C}(\Delta^{n}),{\mathcal{D}})$. See [DS1] for more information, as well for a very detailed description of the functor $\mathfrak{C}$. We call $\mathfrak{C}$ the categorification functor, and $N$ the coherent nerve.

It is useful to also consider the composite of adjoint pairs

where the left adjoints all point left to right. Here $c$ is the functor which regards a category as a discrete simplicial category, and $\pi_{0}$ is the functor which applies the usual $\pi_{0}$ to all the mapping spaces. Note that the composite $Nc$ is the classical nerve functor; as $\pi_{0}\mathfrak{C}$ is its left adjoint, this identifies $\pi_{0}\mathfrak{C}$ as the functor which takes a simplicial set and forms the free category from the $0$-simplices and $1$-simplices, imposing composition relations coming from the $2$-simplices. This functor is denoted $\tau_{1}$ in [J2], but we will always call it $\pi_{0}\mathfrak{C}$ in the present paper.

The following result is due to Joyal:

The proof will be given in Appendix B. The following is an immediate corollary:

Note that Corollary 2.19 adds another equivalent condition to the list given in Proposition 2.2. Also observe that this corollary implies a two-out-of-three property for quasi-isomorphisms: for a $2$-simplex in $X$, if two of its three faces are quasi-isomorphisms then so is the third.

We now also have the following:

Let ${\mathcal{M}}$ be a model category, and let $c{\mathcal{M}}$ be the Reedy model category of cosimplicial objects in ${\mathcal{M}}$ [H, Chapter 15]. For any $X\in{\mathcal{M}}$ we will write $cX$ for the constant cosimplicial object consisting of $X$ in every dimension, where every coface and codegeneracy is the identity.

If $X\in{\mathcal{M}}$, a cosimplicial resolution of $X$ is a Reedy cofibrant replacement $Q^{\bullet}\stackrel{{\scriptstyle\sim}}{{\longrightarrow}}cX$. Given such a cosimplicial resolution and an object $Z\in{\mathcal{M}}$, we may form the simplicial set ${\mathcal{M}}(Q^{\bullet},Z)$ given by

It is known [H, 16.5.5] that if $Z\rightarrow Z^{\prime}$ is a weak equivalence between fibrant objects then the induced map ${\mathcal{M}}(Q^{\bullet},Z)\rightarrow{\mathcal{M}}(Q^{\bullet},Z^{\prime})$ is a Kan equivalence of simplicial sets.

For any object $X\in{\mathcal{M}}$, let ${\mathcal{Q}}(X)$ be the category whose objects are pairs $[Q,Q\rightarrow X]$ where $Q$ is cofibrant and $Q\rightarrow X$ is a weak equivalence. For any object $Y\in{\mathcal{M}}$, there is a functor

sending $[Q,Q\rightarrow X]$ to ${\mathcal{M}}(Q,Y)$. We can regard this functor as taking values in ${s{\mathcal{S}et}}$ by composing with the embedding ${\mathcal{S}et}\hookrightarrow{s{\mathcal{S}et}}$.

Consider the simplicial set $\hocolim_{{\mathcal{Q}}(X)^{op}}{\mathcal{M}}(-,Y)$. We fix our model for the hocolim functor to be the result of first taking the simplicial replacement of a diagram and then applying geometric realization. Notice in our case that in dimension $n$ the simplicial replacement consists of diagrams of weak equivalences $Q_{0}\xleftarrow{\sim}Q_{1}\xleftarrow{\sim}\cdots\xleftarrow{\sim}Q_{n}$ over $X$ (with each $Q_{i}\rightarrow X$ in ${\mathcal{Q}}(X)$), together with a map $Q_{0}\rightarrow Y$. This shows that the simplicial replacement is nothing but the nerve of the category for which an object is a zig-zag $[X\stackrel{{\scriptstyle\sim}}{{\longleftarrow}}Q\rightarrow Y]$, where $Q$ is cofibrant and $Q\rightarrow X$ is a weak equivalence; and a map from $[X\stackrel{{\scriptstyle\sim}}{{\longleftarrow}}Q\rightarrow Y]$ to $[X\stackrel{{\scriptstyle\sim}}{{\longleftarrow}}Q^{\prime}\rightarrow Y]$ is a map $Q^{\prime}\rightarrow Q$ making the evident diagram commute.

Categories of zig-zags like the one considered above were first studied in [DK3]. There are many variations, and it is basically the case that all sensible variations have Kan equivalent nerves; moreover, these nerves are Kan equivalent to the homotopy function complex $\hMap_{\mathcal{M}}(X,Y)$ (defined to be the space of maps in the simplicial localization of ${\mathcal{M}}$ with respect to the weak equivalences). We will next recall some of this machinery. In addition to [DK3], see [D1].

Following [DK3], write $(\Wcofib)^{-1}{\mathcal{M}}(\Wfib)^{-1}(X,Y)$ to denote the category whose objects are zig-zags

and where the maps are natural transformations of diagrams which are the identity on $X$ and on $Y$. Similarly, let $W^{-1}{\mathcal{M}}W^{-1}(X,Y)$ be the category whose objects are zig-zags

let ${\mathcal{M}}(\Wfib)^{-1}{\mathcal{M}}(X,Y)$ be the category whose objects are zig-zags

and so on.

Note that there are natural inclusions of two types: an example of the first is ${\mathcal{M}}(\Wfib)^{-1}(X,Y)\hookrightarrow{\mathcal{M}}\W^{-1}(X,Y)$ (induced by $\Wfib\hookrightarrow\W$), and an example of the second is

which sends

The following proposition is a very basic one in this theory, and will be used often in the remainder of the paper; we have included the proof for completeness, and because it is simple.

Let $QX^{\bullet}\rightarrow X$ be a cosimplicial resolution of $X$ in $c{\mathcal{M}}$. Following [DK3], we now relate the simplicial set ${\mathcal{M}}(QX^{\bullet},Y)$ to the nerves of the categories of zig-zags considered above.

For any simplicial set $K$, let ${\bf\Delta}K$ be the category of simplices of $K$. This is none other than the overcategory $(S\downarrow K)$, where $S\colon{\bf\Delta}\rightarrow{s{\mathcal{S}et}}$ is the functor $[n]\mapsto\Delta^{n}$. It is known that the nerve of ${\bf\Delta}K$ is naturally Kan equivalent to $K$ (see [D1, text prior to Prop. 2.4] for an explanation).

There is a functor ${\bf\Delta}{\mathcal{M}}(QX^{\bullet},Y)\rightarrow{\mathcal{M}}(\Wfib)^{-1}(X,Y)$ sending $([n],QX^{n}\rightarrow Y)$ to $[X\stackrel{{\scriptstyle{\sim}}}{{\twoheadleftarrow\joinrel\relbar}}QX^{n}\longrightarrow Y]$.

Recall from Section 2 that $E\colon{\mathcal{S}et}\rightarrow{s{\mathcal{S}et}}$ denotes the 0-coskeleton functor. For a set $S$, we may also describe $ES$ as the nerve of the groupoid $E_{G}S$ with object set $S$ and a single morphism $a\rightarrow b$ for each $a,b\in S$.

Recall the Reedy model structure [H, Chapter 15.3] on the category of cosimplicial objects in a model category. In the present section, the Reedy structure we will use will always be on $c({s{\mathcal{S}et}}_{J})$, not $c({s{\mathcal{S}et}}_{K})$. However, note that any Reedy-Joyal equivalence is also a Reedy-Kan equivalence.

The forgetful functor ${\bf\Delta}\hookrightarrow{\mathcal{S}et}$ describes a cosimplicial set whose $n$th object is $[n]=\{0,1,\ldots,n\}$. Applying the functor $E$ gives a cosimplicial object $[n]\mapsto E^{n}=N(E_{G}([n]))$ in ${s{\mathcal{S}et}}_{J}$. It is easy to check that $E^{\bullet}$ is Reedy cofibrant in $c({s{\mathcal{S}et}}_{J})$, and the above lemma shows that each $E^{n}$ is contractible in ${s{\mathcal{S}et}}_{J}$. Note that the evident inclusion of categories $[n]\hookrightarrow E_{G}([n])$ induces a levelwise cofibration $\Delta^{\bullet}\hookrightarrow E^{\bullet}$, which in each level is a Kan equivalence but not a Joyal equivalence.

For any simplicial set $X$, the cosimplicial object $[n]\mapsto X\times E^{n}$ is a cosimplicial resolution of $X$ with respect to the model structure ${s{\mathcal{S}et}}_{J}$.

In his book, Lurie at various times uses three internal models for the mapping space between vertices in a simplicial set $S$. They are called $\Hom^{R}_{S}$, $\Hom^{L}_{S}$, and $\Hom_{S}$; the descriptions of all of these can be found in [L, Section 1.2.2]. In this subsection we show that each of these can be understood as the mapping space coming from a cosimplicial resolution of $\Delta^{1}$. We also give one new model, $\Hom^{E}_{S}$, which will be very useful later. See Remark 4.8 for more about the Lurie models.

Recall that for any simplicial sets $M$ and $N$, the join $M\star N$ is a simplicial set with

where we put $M_{-1}=N_{-1}=\Delta^{0}$ (see [L, 1.2.8.1]). Note that $\star$ is a bifunctor and that $M\star\emptyset=\emptyset\star M=M$, so in particular there are natural inclusions $M\hookrightarrow M\star N$ and $N\hookrightarrow M\star N$. Note as well that $M\star\Delta^{0}$ and $\Delta^{0}\star M$ are cones on $M$, and that $\Delta^{n}\star\Delta^{r}\cong\Delta^{n+r+1}$.

We let $C_{R}(M)$ and $C_{L}(M)$ denote the quotient $(M\star\Delta^{0})/M$ and $(\Delta^{0}\star M)/M$, respectively. For any simplicial set $M$, let $C_{cyl}(M)$ be the pushout

Note that $C_{R}(M),C_{L}(M)$, and $C_{cyl}(M)$ each has exactly two $0$-simplices and comes with a natural map to $\Delta^{1}$ (which is unique over $\partial\Delta^{1}$).