nLab
simplicial localization

See also derived hom space

Idea

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

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

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

$${\mathrm{Ho}}_C(a,b)\simeq {\Pi }_0(LC(a,b))$$Ho_C(a,b) \simeq \Pi_0 (L C(a,b))

(where ${\Pi }_0$ gives the 0th simplicial homotopy groupoid).

If the homotopical structure on $C$ extends to that of a (combinatorial) model category, then there is another procedure to obtain a simplicially enriched category from $C$, the (∞,1)-category presented by a combinatorial model category. This $(\mathrm{\infty },1)$-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 $U:\mathrm{Cat}\to \mathrm{Grph}$ be the forgetful functor that sends a (small) category to its underlying reflexive graph, and let $F:\mathrm{Grph}\to \mathrm{Cat}$ be its left adjoint. We then get a comonad $𝔾=(G,ϵ,\delta )$ on $\mathrm{Cat}$, and as usual this defines a functor $G_{•}:\mathrm{Cat}\to [{\Delta }^{\mathrm{op}},\mathrm{Cat}]$ from Cat to simplicial objects in Cat equipped with a canonical augmentation, where $G_{n}C=G^{n+1}C$.

Definition

The standard resolution of a small category $C$ is defined to be the simplicial category $G_{•}C$.

Note that this is also a simplicial category in the strong sense, i.e. $\mathrm{ob}{G}_{•}C$ is discrete! Thus we may also regard $G_{•}C$ as an sSet-category. This is a resolution in the sense that the augmentation $ϵ:G_{•}C\to C$ is a Dwyer-Kan equivalence. (In fact, for objects $X$ and $Y$ in $C$, the morphism $G_{•}C(X,Y)\to C(X,Y)$ admits an extra degeneracy and hence a contracting homotopy.)

Definition

The standard simplicial localization of a relative category $(C,W)$ is the simplicial category $L_{•}(C,W)$ where $L_n(C,W)=G_{n}C[{G_{n}W}^{-1}]$.

This appears as (DwyerKanLocalizations, def. 4.1). Again, $L_{•}C$ is a simplicial category in the strong sense, because $G_{•}C$ is.

Hammock localization

Definition

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

$$L^{H}C(X,Y)\in \mathrm{sSet}$$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 $C$

  $$X\stackrel{\simeq }{\leftarrow }{K}_1\to K_2\stackrel{\simeq }{\leftarrow }{K}_3\to \cdots \to Y,$$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 $W$;

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

  $$\begin{array}{ccccccc}& & K_1& \to & K_2& \stackrel{\simeq }{\leftarrow }& \cdots \\ & {}^{\simeq }\swarrow & & & & & & & {\searrow }^{}\\ X& & {\downarrow }^{\simeq }& & {\downarrow }^{\simeq }& \cdots & & & & Y\\ & {}_{\simeq }\nwarrow & & & & & & & {\nearrow }^{}\\ & & L_1& \to & L_2& \stackrel{\simeq }{\leftarrow }& \cdots \end{array}$$\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 $X,Y,Z$ three objects, there is an evident compositing morphism

$$L^{H}C(X,Y)\times L^{H}C(Y,Z)\to L^{H}C(X,Z)$$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 $L^{H}C$ obtained this way is the hammock localization of $(C,W)$.

This appears as (DwyerKanCalculating, def. 2.1).

Properties

Basic properties

Proposition

For $(C,W)$ a category with weak equivalences, write $L^{H}C\in \mathrm{sSet}\mathrm{Cat}$ for its hammock localization and $C[W^{-1}]\in \mathrm{Cat}$ for its ordinary localization. Write $\mathrm{Ho}(L^{H}C)\in \mathrm{Cat}$ for the category with the same objects as $C$ and morphisms between $X$ and $Y$ being ${\pi }_0{L}^{H}C(X,Y)$.

There is an equivalence of categories

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

This appears as (DwyerKanCalculating, prop. 3.1).

Proposition

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

$$(f:X\to Y)\in W\subset \mathrm{Mor}(C)$$(f : X \to Y) \in W \subset Mor(C)

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

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

and

$$f^{*}:L^{H}C(Y,U)\stackrel{\simeq }{\to }{L}^{H}C(X,U).$$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 $(\mathrm{\infty },1)$-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 $(C,W)$ and $(C\prime ,W\prime )$ be categories with weak equivalences. Write $L^{H}C,L^{H}C\prime \in \mathrm{sSet}\mathrm{Cat}$ for the corresponding hammock localizations.

Then for $F_1,F_2:C\to C\prime$ two homotopical functors (functors respecting the weak equivalences, i.e. $F_i(W)\subset W\prime$) with

$$\eta :F_1\Rightarrow F_2$$\eta : F_1 \Rightarrow F_2

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

$$\begin{array}{ccc}& & L^{H}C\prime (F_1(X),F_1(Y))\\ & {}^{L^H{F}_1}\nearrow & & {\searrow }^{\eta (Y{)}_{*}}\\ L^{H}C(X,Y)& & \Downarrow & & L^{H}C\prime (F_1(X),F_2(Y))\\ & {}_{L^H{F}_2}\searrow & & {\nearrow }_{\eta (X{)}^{*}}\\ & & L^{H}C\prime (F_2(X),F_2(Y))\end{array}.$$\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 $i:(C_1,W_1)\hookrightarrow (C_2,W_2)$ be a full subcategory such that

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

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

   $$i\circ Q\Rightarrow {\mathrm{Id}}_{C_2}.$$i \circ Q \Rightarrow Id_{C_2} \,.

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

$$L^H{C}_1\simeq L^H{C}_2.$$L^H C_1 \simeq L^H C_2 \,.

Proof

We have to check that $i$ 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 $C$ be a simplicial model category. Write $C^{\circ }$ for the full $\mathrm{Set}$-subcategory on the fibrant and cofibrant objects.

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

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

References

The original articles are

• William Dwyer, Dan Kan,

  Simplicial localizations of categories , J. Pure Appl. Algebra 17 (1980), 267–284. (pdf)

  Calculating simplicial localizations , J. Pure Appl. Algebra 18 (1980), 17–35. (pdf)

  Function complexes in homotopical algebra , Topology 19 (1980), 427–440.

  Equivalences between homotopy theories of diagrams , Algebraic topologx and algebraic K-theory, (Princeton, N.J. 1983) , Ann. of Math. Stud. 113, Princeton University Press, Princeton, N.J. 1987 .

• William Dwyer, Philip Hirschhorn, Daniel Kan, Jeff Smith, Homotopy Limit Functors on Model Categories and Homotopical Categories , volume 113 of Mathematical Surveys and Monographs

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

• Tim Porter, $S$-Categories, $S$-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:

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