nLab
(infinity,1)-categorical hom-space

Idea

There are several ways to model an (∞,1)-category $C$ by an ordinary category $C$ equipped with some extra structure: for instance $C$ may be a category with weak equivalences or a model category. In all of these models, given two objects $X,Y\in C$, there is a way to construct an ∞-groupoid $C(X,Y)$ that is the correct hom-object of the (∞,1)-category $C$ – this is the $(\mathrm{\infty }\mathrm{,1})$-categorical hom-space modeled by $C$, often called the derived hom space and then denoted $R\mathrm{Hom}(X,Y)$.

There are various equivalent explicit expressions for $R\mathrm{Hom}$. These are described and compared in the following.

I don't like this term, (for reasons other than using the word ‘categorical’ ^_^). Is there any hom-space that is not $(\mathrm{\infty }\mathrm{,1})$-categorial? It seems to me that if your hom-objects are spaces, then you're an $(\mathrm{\infty }\mathrm{,1})$-category; and if you're an $(\mathrm{\infty }\mathrm{,1})$-category, then your hom-objects could be any spaces. So how about hom-space? —Toby

Urs Schreiber: I see your point. On the other hand probably few readers will expect behind a title “hom-space” more than a remark about the definition of a Top-enriched category. Here the point is the construction of these from 1-categorical data.

How can we say this better?

Toby: Well, I also thought of ‘hom-$\mathrm{\infty }$-groupoid’ and ‘hom-simplicial set’ (at which point the hyphen in ‘hom-object’ looks worse and worse), but these struck me as too specific, although they match what you're saying here.

Urs Schreiber: Choices like “hom-simplicial set” or similar still isn’t good enough for what this entry is about, though: for instance in a simplicially enriched model category all hom-objects are of course “hom-simplicial sets”, but the right ”$(\mathrm{\infty }\mathrm{,1})$-categori(c)al hom space” between two objects is the hom-simplicial set between a fibrant-cofibrant replacement of these two objects.

Other people would say “derived hom space”. I reworded the above introduction and mention that term now.

Toby: I could go for ‘derived hom-space’ (with or without hyphen, although we've been using hyphens here for some reason). That clarifies that it's not the naïve hom-space but doesn't imply that it's the $\mathrm{\infty }$-categorification of a $1$-categorial hom-space.

Urs: At least “derived hom-space” has the advantage that it follows wide-spread practice. However, to me this wide-spread practice looks like a bad practice: the term “derived” is used widely but inconsistently and unsystematically. And it carries with it the historical baggage of its origin as a procedure whose conceptual meaning wasn’t understood. “Derived” essentially just indicates: “there is some procedure that gives us this construction from given data”, never mind if we know what the procedure actually means. Today we know what it actually means: it is all about constructing $(\mathrm{\infty }\mathrm{,1})$-categories. I feel on the $n$Lab we should play Bourbaki and implement good terminology from the point of view of higher category theory.

Toby: I guess that I would pick ‘hom-$\mathrm{\infty }$-groupoid’ (or else ‘hom-$(\mathrm{\infty }\mathrm{,0})$-category’) with that goal in mind. Why say ‘space’ at all if we want to invoke higher categories?

Urs: true.

Interrelation between the different constructions

For $(C,W\subset \mathrm{Mor}(C))$ a category with weak equivalences, Dwyer-Kan simplicial localization produces an SSet-enriched category as follows

For $X,Y\in \mathrm{Obj}(C)$ and for $n\in \mathbb{N}$ define a category $w{\mathrm{Mor}}_C^n(X,Y)$

• whose objects are zig-zags of morphisms in $C$ of length $n$

  $$X=X_0\leftarrow X_1\to X_2\leftarrow \cdots \to X_{n-1}\leftarrow X_n=Y$$X = X_0 \leftarrow X_1 \to X_2 \leftarrow \cdots \to X_{n-1} \leftarrow X_n = Y

  such that each morphism going to the left, $X_{2k}\leftarrow X_{2k+1}$, is a weak equivalence, an element in $W$.

• morphisms between such objects $(X,X_i,Y)\to (X\prime ,X{\prime }_i,Y\prime )$ are collections of weak equivalences $(X_i\to X{\prime }_i)$ for all $0<i<n$ such that all triangles and squares commute.

Write $N({\mathrm{wMor}}_C^n(X,Y))$ for the nerve of this category, a simplicial set.

Then the hammock localization $L^{H}C$ of $C$ is the simplicially enriched category with objects those of $C$ and hom-objects given by the colimit over the length of these hammock hom-categories

$$L^{H}C(X,Y)={\mathrm{colim}}_{n}N({\mathrm{wMor}}_C^n(X,Y)).$$L^H C(X,Y) = colim_n N(wMor_C^n(X,Y)) \,.

The Kan fibrant replacement of this simplicial set is the hom-space between $X$ and $Y$ of the $(\mathrm{\infty }\mathrm{,1})$-category modeled by $(C,W)$.

If the structure of a category with weak equivalences is enhanced to that of a model category, then there are several other ways to construct hom-simplicial sets. The following diagram shows how all these different ways are related and equivalent to the Dwyer-Kan cnstruction $L^{H}C(X,Y)$ – while being usually way more tractable.

Suppose now that $Q:C\to C$ is a cofibrant replacement functor? and $R:C\to C$ a fibrant replacement functor?, ${\Gamma }^{•}:C\to (\mathrm{cC}{)}_c$ a cosimplicial resolution functor? and ${\Lambda }_{•}:C\to (\mathrm{sC}{)}_f$ a simplicial resolution functor? in the model category $C$.

Theorem (Dwyer–Kan)

There are natural weak equivalences between the following equivalent realizations of this SSet hom-object:

$$\begin{array}{ccccc}{\mathrm{Mor}}_C({\Gamma }^{•}X,RY)& & & & {\mathrm{Mor}}_C(QX,{\Lambda }_{•}Y)\\ & {\searrow }^{\simeq }& & {}^{\simeq }\swarrow \\ & & \mathrm{diag}{\mathrm{Mor}}_C({\Gamma }^{•}X,{\Lambda }_{•}Y)\\ & & {\uparrow }^{\simeq }\\ & & {\mathrm{hocolim}}_{p,q\in {\Delta }^{\mathrm{op}}\times {\Delta }^{\mathrm{op}}}{\mathrm{Mor}}_C({\Gamma }^{p}X,{\Lambda }_{q}Y)\\ & & {\downarrow }^{\simeq }\\ & & N{\mathrm{wMor}}_C^3(X,Y)\\ & & {\downarrow }^{\simeq }\\ & & {\mathrm{Mor}}_{L^{H}C}(X,Y)\end{array}.$$\array{ Mor_C(\Gamma^\bullet X, R Y) &&&& Mor_C(Q X, \Lambda_\bullet Y) \\ & \searrow^\simeq && {}^\simeq \swarrow \\ && diag Mor_C(\Gamma^\bullet X, \Lambda_\bullet Y) \\ && \uparrow^\simeq \\ && hocolim_{p,q \in \Delta^{op} \times \Delta^{op}} Mor_C(\Gamma^p X, \Lambda_q Y) \\ &&\downarrow^\simeq \\ &&N wMor_C^3(X,Y) \\ &&\downarrow^\simeq \\ &&Mor_{L^H C}(X,Y) } \,.

Details

Enriched homs between cofibrant/fibrant objects

We describe here in more detail properties of hom-objects in a simplicial model category for the case that the domain objects are cofibrant and the codomain objects are fibrant.

The crucial axiom used for this is the axiom of an enriched model category $C$ which says that

• the tensor operation

  $$\cdot :C\times \mathrm{SSet}\to C$$\cdot : C \times SSet \to C

  is a Quillen bifunctor;

• or equivalently that for $X\to Y$ a cofibration and $A\to B$ a fibration the induced morphism

  $$C(Y,A)\to C(X,A){\times }_{C(X,B)}C(Y,B)$$C(Y, A) \to C(X,A) \times_{C(X,B)} C(Y,B)

  is a fibration, which is acyclic if either $X\to Y$ or $A\to B$ is.

First of all the first statement directly implies that for $\varnothing \in C$ the initial object and $A\in C$ any object, the simplicial set $C(\varnothing ,A)=*$ is the terminal simplicial set, because for any simplicial set $S$

$$\begin{array}{rl}\mathrm{SSet}(S,C(\varnothing ,A))& ={\mathrm{Hom}}_C(\varnothing \cdot S,A)\\ & ={\mathrm{Hom}}_C({\mathrm{colim}}_{\varnothing }\cdot S,A)\\ & ={\mathrm{Hom}}_C(\varnothing ,A)\\ & =*\end{array},$$\begin{aligned} SSet(S,C(\emptyset, A)) & = Hom_C(\emptyset \cdot S, A) \\ & = Hom_C(colim_{\emptyset} \cdot S, A) \\ & = Hom_C(\emptyset, A) \\ &= {*} \end{aligned} \,,

where we use that the tensor Quillen bifunctor is required to respect colimits and that the empty colimit is the initial object. (All equality signs here denote isomorphisms, to distinguish them from weak equivalences.)

Similarly one has for all $X$ that $C(X,*)=*$.

Using this, the second equivalent form of the enrichment axiom has as a special case the following statement.

Lemma

In a simplicial model category $C$, for $X\in C$ cofibrant and $A\in C$ fibrant, the simplicial set $C(X,A)$ is a Kan complex.

Proof

We apply the enriched model category axiom to the cofibration $\varnothing \to X$ and the fibration $A\to *$ to obtain a fibration

$$C(X,A)\to C(\varnothing ,A){\times }_{C(\varnothing ,*)}C(X,*).$$C(X,A) \to C(\emptyset, A) \times_{C(\emptyset,{*})} C(X,{*}) \,.

The right hand is the pullback of the terminal simplicial set $*={\Delta }^0$ to itself, hence is itself the point. So we have a fibration $C(X,A)\to *$ and $C(X,A)$ is a fibrant object in the standard model structure on simplicial sets, hence a Kan complex. .

Lemma

In a simplicial model category $C$, for $X\in C$ cofibrant and $f:A\to B$ a fibration, the morphism of simplicial sets $C(X,f):C(X,A)\to C(X,B)$ is a Kan fibration that is a weak homotopy equivalence if $f$ is acyclic.

Dually, for $i:X\to Y$ a cofibration and $A$ fibrant, the morphism $C(i,A):C(X,A)\to C(Y,A)$ is a cofibration of simplicial sets.

Proof

This is as before. Explicity, consider the first case, the second one is the formal dual of that:

We enter the enrichment axiom with the morphisms $\varnothing \to X$ and $A\to B$ and find for the required pullback that

$$C(\varnothing ,A){\times }_{C(\varnothing ,B)}C(X,B)=*{\times }_{*}C(X,B)=C(X,B)$$C(\emptyset,A) \times_{C(\emptyset, B)} C(X,B) = {*} \times_{*} C(X,B) = C(X,B)

and hence that $C(X,A)\to C(X,B)$ is, indeed, a fibration, which is acyclic if $A\to B$ is.

Proposition

Let $C$ be a simplicial model category.

Then for $X$ a cofibant object and

$$f:A\stackrel{\simeq }{\to }B$$f : A \stackrel{\simeq}{\to} B

a weak equivalence between fibrant objects, the enriched hom-functor

$$C(X,f):C(X,A)\to C(X,B)$$C(X,f) : C(X,A) \to C(X,B)

is a weak homotopy equivalence of Kan complexes.

Similarly, for $A$ a fibrant object and $j:X\stackrel{\simeq }{\to }Y$ a weak equivalence between cofibrant objects, the morphism

$$C(j,A):C(X,A)\to C(Y,A)$$C(j,A) : C(X,A) \to C(Y,A)

is a weak homotopy equivalence of Kan complexes.

Proof

The second case is formally dual to the first, so we restrict attention to the first one.

By the above, the axioms of an enriched model category ensure that the above statement is true when $f$ is in addition a fibration. So we reduce the situation to that case.

This is possible because both $A$ and $B$ are assumed to be fibrant. This allows to apply the factorization lemma that is described in great detail at category of fibrant objects. By this lemma, for every morphism $f:A\to B$ between fibrant objects there is a commutative diagram

$$\begin{array}{ccc}& & E_{f}B\\ & {}^{\in \mathrm{fib}\cap W}\swarrow & & {\searrow }^{\in \mathrm{fib}}\\ A& & \stackrel{\simeq }{\to }& & B\end{array}$$\array{ && \mathbf{E}_f B \\ & {}^{\mathllap{\in fib \cap W}}\swarrow && \searrow^{\mathrlap{\in fib}} \\ A &&\stackrel{\simeq}{\to}&& B }

Since $f$ is assumed a weak equivalence it follows by 2-out-of-3 that $E_{f}B$ is also a weak equivalence.

Therefore by the above properties of simpliciall enriched categories we obtain a span of acyclic fibrations of Kan complexes

$$C(X,A)\stackrel{\simeq }{\leftarrow }C(X,E_{f}B)\stackrel{\simeq }{\to }C(X,B).$$C(X,A) \stackrel{\simeq}{\leftarrow} C(X, \mathbf{E}_f B) \stackrel{\simeq}{\to} C(X,B) \,.

By the Whitehead theorem every weak equivalence of Kan complexes is a homotopy equivalence, hence there is a weak equivalence

$$C(X,A)\stackrel{\simeq }{\to }C(X,E_{f}B)\stackrel{\simeq }{\to }C(X,B)$$C(X,A) \stackrel{\simeq}{\to} C(X,\mathbf{E}_f B) \stackrel{\simeq}{\to} C(X,B)

that is homotopic to our $C(X,f)$. Therefore this is also a weak equivalence.

References

A useful quick review of the interrelation of the various constructions of derived hom spaces is page 14, 15 of

• Clark Barwick, On (enriched) left Bousfield localization of model categories (arXiv)

where the above diagram is taken from.

The definition in terms of simplicial and fibrant/cofibrant resolutions is described in detail in sections 16, 17 of

• Hirschhorn, Model categories and their localization .