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(infinity,1)-categorical hom-space

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Homotopy theory

(,1)(\infty,1)-Category theory

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model category

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Presentation of (,1)(\infty,1)-categories

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Mapping space

Contents

Idea

Where an ordinary category has a hom-set, an (∞,1)-category has an ∞-groupoid of morphisms between any two objects, a hom-space.

There are several ways to present an (∞,1)-category C\mathbf{C} by an ordinary category CC equipped with some extra structure: for instance CC may be a category with weak equivalences or a model category or even a simplicial model category. In all of these presentations, given two objects X,YCX, Y \in C, there is a way to construct a simplicial sets C(X,Y)\mathbb{R}\mathbf{C}(X,Y) that presents the hom-∞-groupoid C(X,Y)\mathbf{C}(X,Y). This simplicial set – or rather its homotopy type – is called the derived hom space or homotopy function complex and denoted RHom(X,Y)\mathbf{R}Hom(X,Y) or similarly.

Presentations

There are many ways to present an (∞,1)-category by category theoretic data, and for each of these there are corresponding tools for explicitly computing the derived hom spaces.

The most basic data is that of a category with weak equivalences. Here the derived hom spaces can be constructed in terms of zig-zags of morphisms by a process called simplicial localization. This we discuss below in For a category with weak equivalences.

Particularly useful extra structure on a category with weak equivalences that helps with computing the derived hom spaces is the structure of a model category. Using this one can construc simplicial resolutions of objects – called framings – that generalize cylinder objects and path objects, and then construct the derived hom spaces in terms of direct morphisms between these resolutions. This we discuss below in For a model category.

Still a bit more helpful structure on top of a bare model category is that of a simplicial model category. Here, after a choice of cofibrant and fibrant resolutions of opjects, the derived hom spaces are given already by the sSet-hom objects. This we discuss below in For a simplicial model category.

For a category with weak equivalences

Let (C,WMor(C))(C,W \subset Mor(C)) be a category with weak equivalences.

Definition

Fix nn \in \mathbb{N}. For X,YObj(C)X,Y \in Obj(C), define a category wMor C n(X,Y)wMor_C^n(X,Y)

  • whose objects are zig-zags of morphisms in CC of length nn

    X=X 0X 1X 2X n1X 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 2kX 2k+1X_{2k}\leftarrow X_{2k +1}, is a weak equivalence, an element in WW;

  • morphisms between such objects (X,X i,Y)(X,X i,Y)(X,X_i,Y) \to (X',X'_i,Y') are collections of weak equivalences (X iX i)(X_i \to X'_i) for all 0<i<n0 \lt i \lt n such that all triangles and squares commute.

Definition

Write N(wMor C n(X,Y))N(wMor_C^n(X,Y)) for the nerve of this category, a simplicial set.

The hammock localization L W HCL_W^H C of CC is the simplicially enriched category with objects those of CC and hom-objects given by the colimit over the length of these hammock hom-categories

L HC(X,Y):=lim nN(wMor C n(X,Y)). L^H C(X,Y) := \lim_{\to_n} N(wMor_C^n(X,Y)) \,.

The Kan fibrant replacement of this simplicial set is the derived hom-space between XX and YY of the (,1)(\infty,1)-category modeled by (C,W)(C,W).

For a model category

The derived hom spaces of a model category CC may always be computed in terms of simplicial resolutions with respect to the Reedy model structure [Δ op,C] Reedy[\Delta^{op}, C]_{Reedy}. These resolutions are often called framings (Hovey). These constructions are originally due to (Dwyer-Hirschhorn-Kan).

Let CC be any model category.

Observation

There is an adjoint triple

(constev 0() × ):C() × ev 0const,[Δ op,C], (const \dashv ev_0 \dashv (-)^{\times^\bullet}) : C \stackrel{\overset{const}{\to}}{\stackrel{\overset{ev_0}{\leftarrow}}{\underset{(-)^{\times^\bullet}}{\to}}} \,, [\Delta^{op}, C] \,,

where

  1. constX:[n]Xconst X : [n] \mapsto X;

  2. ev 0X =X 0ev_0 X_\bullet = X_0;

  3. X × :[n]X × nX^{\times^\bullet} : [n] \mapsto X^{\times^n}.

Remark

For XCX \in C fibrant, X × X^{\times^\bullet} is fibrant in the Reedy model structure [Δ op,C] Reedy[\Delta^{op}, C]_{Reedy}.

Proof

The matching morphisms are in fact isomorphisms.

Definition

Let CC be a model category.

  1. For XCX \in C any object, a simplicial frame on XX is a factorization of constXX × const X \to X^{\times^\bullet} into a weak equivalence followed by a fibration in the Reedy model structure [Δ op,C] Reedy[\Delta^{op}, C]_{Reedy}.

  2. A right framing in CC is a functor () :C[Δ op,C](-)_\bullet : C \to [\Delta^{op}, C] with a natural isomorphism (X) 0X(X)_0 \simeq X such that X X_\bullet is a simplicial frame on XX.

Dually for cosimplicial frames.

This appears as (Hovey, def. 5.2.7).

Remark

By remark 1 a simplicial frame X X_\bullet in the above is in particular fibrant in [Δ op,C] Reedy[\Delta^{op}, C]_{Reedy}.

Proposition

For XCX \in C cofibrant and ACA \in C fibrant, there are weak equivalences in sSet QuillensSet_{Quillen}

Hom C(X ,A)diagHom C(X ,A )Hom C(X,A ), Hom_C(X^\bullet, A) \stackrel{\simeq}{\to} diag Hom_C(X^\bullet, A_\bullet) \stackrel{\simeq}{\leftarrow} Hom_C(X, A_\bullet) \,,

(where in the middle we have the diagonal of the bisimplicial set Hom(X ,A )Hom(X^\bullet, A_\bullet)).

This appears as (Hovey, prop. 5.4.7).

Either of these simplicial sets is a model for the derived hom-space Hom(X,A)\mathbb{R}Hom(X,A).

Remark

By developing these constructions further, one obtains a canonical simplicial model category-resolution of (left proper and combinatorial) model categories CC, such that the simplicial resolutions given by framings are just the cofibrant\tofibrant sSetsSet-hom objects as discussed below.

This is discussed at Simplicial Quillen equivalent models.

Proposition

Let CC be a model category, let c wC\mathrm{c}_\mathrm{w} C be the full subcategory of [Δ,C][\Delta, C] spanned by the cosimplicial objects whose coface and codegeneracy operators are weak equivalences, and let s wC\mathrm{s}_\mathrm{w} C be the full subcategory of [Δ op,C][\Delta^{op}, C] spanned by the simplicial objects whose face and degeneracy operators are weak equivalences.

  1. const:Cc wCconst : C \to \mathrm{c}_\mathrm{w} C is the right half of an adjoint homotopical equivalence of homotopical categories, and const:Cs wCconst : C \to \mathrm{s}_\mathrm{w} C is the left half of an adjoint homotopical equivalence of homotopical categories.
  2. The functor diagHom C:(c wC) op×s wCsSet\operatorname{diag} Hom_C : (\mathrm{c}_\mathrm{w} C)^{op} \times \mathrm{s}_\mathrm{w} C \to sSet admits a right derived functor.
  3. The induced functor (HoC) op×HoCHosSet(\operatorname{Ho} C)^{op} \times \operatorname{Ho} C \to \operatorname{Ho} sSet is the derived hom-space functor.

For a simplicial model category

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 CC which says that

  • the tensor operation

    :C×SSetC \cdot : C \times SSet \to C

    is a Quillen bifunctor;

  • or equivalently that for XYX \to Y a cofibration and ABA \to B a fibration the induced morphism

    C(Y,A)C(X,A)× 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 XYX \to Y or ABA \to B is.

First of all the first statement directly implies that for C\emptyset \in C the initial object and ACA \in C any object, the simplicial set C(,A)=*C(\emptyset,A) = {*} is the terminal simplicial set, because for any simplicial set SS

SSet(S,C(,A)) =Hom C(S,A) =Hom C(colim S,A) =Hom C(,A) =*, \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 XX that C(X,*)=*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 CC, for XCX \in C cofibrant and ACA \in C fibrant, the simplicial set C(X,A)C(X,A) is a Kan complex.

Proof

We apply the enriched model category axiom to the cofibration X\emptyset \to X and the fibration A*A \to {*} to obtain a fibration

C(X,A)C(,A)× C(,*)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 *=Δ 0{*} = \Delta^0 to itself, hence is itself the point. So we have a fibration C(X,A)* C(X,A) \to {*} and C(X,A)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 CC, for XCX \in C cofibrant and f:ABf : A \to B a fibration, the morphism of simplicial sets C(X,f):C(X,A)C(X,B)C(X,f) : C(X,A) \to C(X,B) is a Kan fibration that is a weak homotopy equivalence if ff is acyclic.

Dually, for i:XYi : X \to Y a cofibration and AA fibrant, the morphism C(i,A):C(X,A)C(Y,A)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 X\emptyset \to X and ABA \to B and find for the required pullback that

C(,A)× C(,B)C(X,B)=*× *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)C(X,B)C(X,A) \to C(X,B) is, indeed, a fibration, which is acyclic if ABA \to B is.

Proposition

Let CC be a simplicial model category.

Then for XX a cofibant object and

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

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

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

is a weak homotopy equivalence of Kan complexes.

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

C(j,A):C(X,A)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 ff is in addition a fibration. So we reduce the situation to that case.

This is possible because both AA and BB 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:ABf : A \to B between fibrant objects there is a commutative diagram

E fB fibW fib A B \array{ && \mathbf{E}_f B \\ & {}^{\mathllap{\in fib \cap W}}\swarrow && \searrow^{\mathrlap{\in fib}} \\ A &&\stackrel{\simeq}{\to}&& B }

Since ff is assumed a weak equivalence it follows by 2-out-of-3 that E fB\mathbf{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)C(X,E fB)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)C(X,E fB)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)C(X,f). Therefore this is also a weak equivalence.

Comparison

Let CC be a model category. We discuss how its simplicial function complexes from prop. 2 are related to the simplicial localization from def. 1 and def. 2.

Suppose now that Q:CCQ : C \to C is a cofibrant replacement functor and R:CCR : C \to C a fibrant replacement functor, Γ :C(cC) c\Gamma^\bullet : C \to (cC)_c a cosimplicial resolution functor and Λ :C(sC) f\Lambda_\bullet : C \to (sC)_f a simplicial resolution functor in the model category CC.

Theorem

(Dwyer–Kan)

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

Mor C(Γ X,RY) diagMor C(Γ X,Λ Y) Mor C(QX,Λ Y) hocolim p,qΔ op×Δ opMor C(Γ pX,Λ qY) NwMor C 3(X,Y) Mor L HC(X,Y). \array{ Mor_C(\Gamma^\bullet X, R Y) &\stackrel{\simeq}{\to}& diag Mor_C(\Gamma^\bullet X, \Lambda_\bullet Y) &\stackrel{\simeq}{\leftarrow}& Mor_C(Q 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) } \,.

The top row weak equivalences are those of prop. 2

In a category of fibrant objects

There is also an explicit simplicial construction of the derived hom spaces for a homotopical category that is equipped with the structure of a category of fibrant objects. This is described in (Cisinksi 10) and (Nikolaus-Schreiber-Stevenson 12, section 3.6.2).

Properties

Hom-spaces of equivalences

Theorem

For CC a simplicial model category and XX an object, the delooping of the automorphism ∞-group

Aut W(X)Hom(X,X) Aut_W(X) \subset \mathbb{R}Hom(X,X)

has the homotopy type of the component on XX of the nerve N(C W)N(C_W) of the subcategory of weak equivalences:

BAut W(X)N(C W) X. \mathbf{B} Aut_W(X) \simeq N(C_W)_X \,.

The equivalence is given by a finite sequence of zig-zags and is natural with respect to sSet-enriched functors of simplicial model categories that preserve weak equivalences and send a fibrant cofibrant model for XX again to a fibrant cofibrant object.

This is Dwyer-Kan 84, 2.3, 2.4.

Corollary

For CC a model category, the simplicial set N(C W)N(C_W) is a model for the core of the (∞,1)-category determined by CC.

Proof

That core, like every ∞-groupoid is equivalent to the disjoint union over its connected components of the deloopings of the automorphism \infty-groups of any representatives in each connected component.

homotopycohomologyhomology
[S n,][S^n,-][,A][-,A]()A(-) \otimes A
category theorycovariant homcontravariant homtensor product
homological algebraExtExtTor
enriched category theoryendendcoend
homotopy theoryderived hom space Hom(S n,)\mathbb{R}Hom(S^n,-)cocycles Hom(,A)\mathbb{R}Hom(-,A)derived tensor product () 𝕃A(-) \otimes^{\mathbb{L}} A

References

For some original references by William Dwyer and Dan Kan see simplicial localization. For instance

  • William Dwyer, Dan Kan, A classication theorem for diagrams of simplicial sets, Topology 23 (1984), 139-155.

The theory of framings is due to

and in parallel section 5 of

and in sections 16, 17 of

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

Discussion of derived hom spaces for categories of fibrant objects is in

  • Denis-Charles Cisinski, Invariance de la K-théorie par equivalences dérivées, J. K-theory, 6 (2010), 505–546.

and section 3.6.2 of

Revised on May 15, 2014 01:43:17 by Urs Schreiber (77.251.114.72)