nLab
simplicial localization

Context

Locality and descent

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

See also derived hom space

Simplicial localisation

Idea

A category with weak equivalences or homotopical category is a category CC equipped with the information that some of its morphisms, specifically, a subcategory WCore(C)W \supset Core(C), are to be regarded as “weakly invertible”. One way to make this notion precise is through the concept of simplicial localization:

The simplicial localization LCL C of a category CC is an (∞,1)-category realized concretely as a simplicially enriched category which is such that the original category injects into it, CLCC \hookrightarrow L C, such that every morphism in CC that is labeled as a weak equivalence becomes an actual equivalence in the sense of morphisms in (∞,1)-categories in LCL C. And LCL C is in some sense universal with this property.

Passing to the homotopy category of an (∞,1)-category of LCL C then reproduces the homotopy category that can also directly be obtained from CC:

Ho C(a,b)Π 0(LC(a,b)) Ho_C(a,b) \simeq \Pi_0 (L C(a,b))

(where Π 0\Pi_0 gives the 0th simplicial homotopy groupoid).

If the homotopical structure on CC extends to that of a (combinatorial) model category, then there is another procedure to obtain a simplicially enriched category from CC, the (∞,1)-category presented by a combinatorial model category. This (,1)(\infty,1)-category is equivalent to the one obtained by simplicial localization but typically more explicit and more tractable.

See also localization of a simplicial model category.

Construction

See simplicial localization of a homotopical category.

Definition

“Standard” simplicial localization

Let U:CatGrphU : \mathbf{Cat} \to \mathbf{Grph} be the forgetful functor that sends a (small) category to its underlying reflexive graph, and let F:GrphCatF : \mathbf{Grph} \to \mathbf{Cat} be its left adjoint. We then get a comonad 𝔾=(G,ϵ,δ)\mathbb{G} = (G, \epsilon, \delta) on Cat\mathbf{Cat}, and as usual this defines a functor G :Cat[Δ op,Cat]G_\bullet : \mathbf{Cat} \to [\mathbf{\Delta}^{op}, \mathbf{Cat}] from Cat to simplicial objects in Cat equipped with a canonical augmentation, where G nC=G n+1CG_n C = G^{n+1} C.

Definition

The standard resolution of a small category CC is defined to be the simplicial category G CG_\bullet C.

Note that this is also a simplicial category in the strong sense, i.e. obG Cob G_\bullet C is discrete! Thus we may also regard G CG_\bullet C as an sSet-category. This is a resolution in the sense that the augmentation ϵ:G CC\epsilon : G_\bullet C \to C is a Dwyer-Kan equivalence. (In fact, for objects XX and YY in CC, the morphism G C(X,Y)C(X,Y)G_\bullet C (X, Y) \to C (X, Y) admits an extra degeneracy and hence a contracting homotopy.)

Definition

The standard simplicial localization of a relative category (C,W)(C, W) is the simplicial category L (C,W)L_\bullet (C, W) where L n(C,W)=G nC[G nW 1]L_n (C, W) = G_n C [{G_n W}^{-1}].

This appears as (DwyerKanLocalizations, def. 4.1). Again, L CL_\bullet C is a simplicial category in the strong sense, because G CG_\bullet C is.

Hammock localization

Definition

Let (C,W)(C,W) be a category with weak equivalences. For X,YCX,Y \in C any two objects, write

L HC(X,Y)sSet L^H C(X,Y) \in sSet

for the simplicial set which is the nerve of the category defined as follows:

  • objects are equivalence classes of zig-zags of morphisms in CC

    XK 1K 2K 3Y, X \stackrel{\simeq}{\leftarrow} K_1 \to K_2 \stackrel{\simeq}{\leftarrow} K_3 \to \cdots \to Y \,,

    where the left-pointing morphisms are to be in WW;

  • morphisms are equivalence classes of “natural transformations” between such objects, fixing the endpoints:

    K 1 K 2 X Y L 1 L 2 \array{ && K_1 &\to& K_2 &\stackrel{\simeq}{\leftarrow}& \cdots \\ & {}^{\mathllap{\simeq}}\swarrow &&&&& && \searrow^{} \\ X && \downarrow^{\simeq}&& \downarrow^{\simeq}& \cdots&& && Y \\ & {}_{\mathllap{\simeq}}\nwarrow &&&&& && \nearrow^{} \\ && L_1 &\to& L_2 &\stackrel{\simeq}{\leftarrow}& \cdots } \;
  • the equivalence relation identifies two such zig-zags if one is obtained by removing morphisms in one slot and/or by composing morphism that become composable this way.

For X,Y,ZX,Y,Z three objects, there is an evident compositing morphism

L HC(X,Y)×L HC(Y,Z)L HC(X,Z) L^H C(X,Y) \times L^H C(Y,Z) \to L^H C(X,Z)

given by horizontally concatenating hammock diagrams as above.

The simplicially enriched category L HCL^H C obtained this way is the hammock localization of (C,W)(C,W).

This appears as (DwyerKanCalculating, def. 2.1).

Properties

Basic properties

Proposition

For (C,W)(C,W) a category with weak equivalences, write L HCsSetCatL^H C \in sSet Cat for its hammock localization and C[W 1]CatC[W^{-1}] \in Cat for its ordinary localization. Write Ho(L HC)CatHo(L^H C) \in Cat for the category with the same objects as CC and morphisms between XX and YY being π 0L HC(X,Y)\pi_0 L^H C(X,Y).

There is an equivalence of categories

HoL HC(X,Y)C[W 1]. Ho L^H C(X,Y) \simeq C[W^{-1}] \,.

This appears as (DwyerKanCalculating, prop. 3.1).

Proposition

Let (C,W)(C,W) be a category with weak equivalences, and let

(f:XY)WMor(C) (f : X \to Y) \in W \subset Mor(C)

be a weak equivalence. Then for all objects UCU \in C we have that the to concatenation operations on hammocks induce weak homotopy equivalences

f *:L HC(U,X)L HC(U,Y) f_* : L^H C(U,X) \stackrel{\simeq}{\to} L^H C(U,Y)

and

f *:L HC(Y,U)L HC(X,U). f^* : L^H C(Y,U) \stackrel{\simeq}{\to} L^H C(X, U) \,.

This appears as (DwyerKanCalculating, prop. 3.3).

Simplical localization gives all (,1)(\infty,1)-categories

Proposition

Up to Dwyer-Kan equivalence –the weak equivalences in the model structure on sSet-categories – every simplicial category is the simplicial localization of a category with weak equivalences.

This is (DwyerKan 87, 2.5).

If the category with weak equivalences in question happens to carry even the structure of a model category there exist more refined tools for computing the SSet-hom object of the simplicial localization. These are described at (∞,1)-categorical hom-space.

Equivalences between simplicial localizations

Proposition

Let (C,W)(C,W) and (C,W)(C', W') be categories with weak equivalences. Write L HC,L HCsSetCatL^H C, L^H C' \in sSet Cat for the corresponding hammock localizations.

Then for F 1,F 2:CCF_1, F_2 : C \to C' two homotopical functors (functors respecting the weak equivalences, i.e. F i(W)WF_i(W) \subset W') with

η:F 1F 2 \eta : F_1 \Rightarrow F_2

a natural transformation with components in the WW', we have that for all objects X,YCX,Y \in C, there is induced a natural homotopy between morphisms of simplicial sets

L HC(F 1(X),F 1(Y)) L HF 1 η(Y) * L HC(X,Y) L HC(F 1(X),F 2(Y)) L HF 2 η(X) * L HC(F 2(X),F 2(Y)). \array{ && L^H C'(F_1(X), F_1(Y) ) \\ & {}^{\mathllap{L^H F_1}}\nearrow && \searrow^{\mathrlap{\eta(Y)_*}} \\ L^H C(X,Y) && \Downarrow && L^H C'(F_1(X), F_2(Y)) \\ & {}_{\mathllap{L^H F_2}}\searrow && \nearrow_{\mathrlap{\eta(X)^*}} \\ && L^H C' (F_2(X), F_2(Y)) } \,.

This is (DwyerKanComputations, prop. 3.5).

Corollary

Let i:(C 1,W 1)(C 2,W 2)i : (C_1, W_1) \hookrightarrow (C_2, W_2) be a full subcategory such that

  1. ii is homotopy-essentially surjective: for every object c 2C 2c_2 \in C_2 there is an object c 1C 1c_1 \in C_1 and a weak equivalence c 2i(c 1)c_2 \stackrel{\simeq}{\to} i(c_1);

  2. there is a functor Q:(C 2,W 2)(C 1,W 1)Q : (C_2,W_2) \to (C_1, W_1) and a natural transformation

    iQId C 2. i \circ Q \Rightarrow Id_{C_2} \,.

Then we have an equivalence of (∞,1)-categories

L HC 1L HC 2. L^H C_1 \simeq L^H C_2 \,.
Proof

We have to check that ii is an essentially surjective (∞,1)-functor and a full and faithful (∞,1)-functor.

The first condition is immediate from the first assumption. The second follows with prop. 4 (using prop. 2) from the second assumption.

Simplicial localization of model categories

Proposition

Let CC be a simplicial model category. Write C C^\circ for the full SetSet-subcategory on the fibrant and cofibrant objects.

Then C C^\circ and L HCL^H C are connected by an equivalence of (∞,1)-categories.

This is one of the central statements in (DwyerKanFunctionComplexes). The weak homotopy equivalence between C (X,Y)C^\circ(X,Y) and L HC(X,Y)L^H C(X,Y) is in corollary 4.7. The equivalence of \infty-categories stated above follows with this and the diagram of morphisms of simplicial categories in prop. 4.8.

References

The original articles are

A survey of the general topic involved here can be found in the following paper:

  • Tim Porter, SS-Categories, SS-groupoids, Segal categories and quasicategories (arXiv)

Hammock localization is described in Section 4.1 there.

A useful quick collection of facts can be found at the beginning of Section 2 in the following paper:

Revised on July 18, 2013 12:05:41 by Urs Schreiber (82.169.65.155)