simplicial localization

Simplicial localization


Locality and descent

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

See also derived hom space

Simplicial localization


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.


See simplicial localization of a homotopical category.


“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.


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.)


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


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).


Basic properties


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 the connected components 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).


Hammock localization is clearly a functor from the category of relative categories to sSet-enriched categories:

RelativeCatL W HSimplicialCategories RelativeCat \overset{L^H_W}{\longrightarrow} SimplicialCategories

See also Barwick & Kan 12, Sec. 1.5, Spitzweck 10, p. 3.


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

(XfY)WMor(C) (X \overset{f}{\to} Y) \in W \subset Mor(C)

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

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


f *:L HC(Y,U)L HC(X,U). f^* \;\colon\; 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


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


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).


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 \,.

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


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 (Dwyer & Kan 80 FuncComp). 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.


A Quillen equivalence DCD \leftrightarrows C between model categories induces a Dwyer-Kan-equivalence LCLDL C \leftrightarrow L D between their simplicial localizations.

This is made explicit in Mazel-Gee 15, p. 17 to follow from Dwyer & Kan 80 FuncComp, Prop. 4.4 with 5.4.


The original articles:


  • Tim Porter, Section 4.1 of: SS-Categories, SS-groupoids, Segal categories and quasicategories (arXiv)

  • Clark Barwick, Section 2: On (enriched) Bousfield localization of model categories (arXiv)

Further development:

See also:

Last revised on June 12, 2021 at 09:31:03. See the history of this page for a list of all contributions to it.