nLab
simplicial localization

Simplicial localization

Context

Locality and descent

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

See also derived hom space

Simplicial localization

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, one form of which is known as Dwyer–Kan 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 defined as follows. For each natural number nn there is a category defined as follows:

  • its objects are length-nn 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;

  • its morphisms are “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 } \;

L HC(X,Y)L^H C(X,Y) is obtained by

  • taking the coproduct of the nerves of these categories over all nn, and

  • quotienting by the equivalence relation generated by inserting or removing identity morphisms and composing composable morphisms.

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 HCC[W 1]. Ho L^H C \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. (using prop. ) 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:

Last revised on November 17, 2019 at 16:53:05. See the history of this page for a list of all contributions to it.