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 $W \supset Core(C)$, 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 $L C$ 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, $C \hookrightarrow L C$, 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 $L C$. And $L C$ is in some sense universal with this property.

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

$$Ho_C(a,b) \simeq \Pi_0 (L C(a,b))$$

(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 $(\infty,1)$-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 $U : \mathbf{Cat} \to \mathbf{Grph}$ be the forgetful functor that sends a (small) category to its underlying reflexive graph, and let $F : \mathbf{Grph} \to \mathbf{Cat}$ be its left adjoint. We then get a comonad $\mathbb{G} = (G, \epsilon, \delta)$ on $\mathbf{Cat}$, and as usual this defines a functor $G_\bullet : \mathbf{Cat} \to [\mathbf{\Delta}^{op}, \mathbf{Cat}]$ 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 $G_\bullet C$.

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

Definition

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

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

Hammock localization

Definition

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

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

for the simplicial set defined as follows. For each natural number $n$ there is a category defined as follows:

• its objects are length-$n$ zig-zags of morphisms in $C$

  $$X \stackrel{\simeq}{\leftarrow} K_1 \to K_2 \stackrel{\simeq}{\leftarrow} K_3 \to \cdots \to Y \,,$$

  where the left-pointing morphisms are to be in $W$;

• its morphisms are “natural transformations” between such objects, fixing the endpoints:

  $$\array{ && K_1 &\to& K_2 &\stackrel{\simeq}{\leftarrow}& \cdots \\ & {}^{\mathllap{\simeq}}\swarrow &&&&& && \searrow^{} \\ X && \downarrow^{\simeq}&& \downarrow^{\simeq}& \cdots&& && Y \\ & {}_{\mathllap{\simeq}}\nwarrow &&&&& && \nearrow^{} \\ && L_1 &\to& L_2 &\stackrel{\simeq}{\leftarrow}& \cdots } \;$$

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

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

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

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

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

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 $L^H C \in sSet Cat$ for its hammock localization and $C[W^{-1}] \in Cat$ for its ordinary localization. Write $Ho(L^H C) \in Cat$ for the category with the same objects as $C$ and morphisms between $X$ and $Y$ being the connected components $\pi_0 L^H C(X,Y)$.

There is an equivalence of categories

$$Ho L^H C \simeq C[W^{-1}] \,.$$

This appears as (DwyerKanCalculating, prop. 3.1).

Remark

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

$$RelativeCat \overset{L^H_W}{\longrightarrow} SimplicialCategories$$

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

Proposition

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

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

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

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

and

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

This appears as (DwyerKanCalculating, prop. 3.3).

Simplical localization gives all $(\infty,1)$-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 $(C', W')$ be categories with weak equivalences. Write $L^H C, L^H C' \in sSet Cat$ for the corresponding hammock localizations.

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

$$\eta : F_1 \Rightarrow F_2$$

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

$$\array{ && L^H C'(F_1(X), F_1(Y) ) \\ & {}^{\mathllap{L^H F_1}}\nearrow && \searrow^{\mathrlap{\eta(Y)_*}} \\ L^H C(X,Y) && \Downarrow && L^H C'(F_1(X), F_2(Y)) \\ & {}_{\mathllap{L^H F_2}}\searrow && \nearrow_{\mathrlap{\eta(X)^*}} \\ && L^H C' (F_2(X), F_2(Y)) } \,.$$

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

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

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

$$L^H C_1 \simeq L^H C_2 \,.$$

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. (using prop. ) from the second assumption.

Simplicial localization of model categories

Proposition

Let $C$ be a simplicial model category. Write $C^\circ$ for the full $sSet$-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 (Dwyer & Kan 80 FuncComp). The weak homotopy equivalence between $C^\circ(X,Y)$ and $L^H C(X,Y)$ 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.

Proposition

A Quillen equivalence $D \leftrightarrows C$ between model categories induces a Dwyer-Kan-equivalence $L C \leftrightarrow L D$ 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.

Presentation in terms of complete Segal spaces

Using quasicategories as a model of (infinity,1)-categories, there is a construction which computes simplicial localization in terms of complete Segal spaces. See complete Segal spaces#RelationToSimplicialLocalization.

Related concepts

• localizer

• localizing subcategory

References

The original articles:

• William Dwyer, Daniel Kan, Simplicial localizations of categories, J. Pure Appl. Algebra 17 3 (1980), 267-284 [doi:10.1016/0022-4049(80)90049-3]

• William Dwyer, Daniel Kan, Calculating simplicial localizations, J. Pure Appl. Algebra 18 (1980), 17-35 [doi:10.1016/0022-4049(80)90113-9]

• William Dwyer, Daniel Kan, Function complexes in homotopical algebra, Topology 19 (1980), 427-440 [doi:10.1016/0040-9383(80)90025-7, pdf]

• William Dwyer, Daniel Kan, Equivalences between homotopy theories of diagrams, in: Algebraic topology and algebraic K-theory, Ann. of Math. Stud. 113, Princeton University Press (1988) [doi:10.1515/9781400882113-009]

and in modernized form:

• William Dwyer, Philip Hirschhorn, Daniel Kan, Jeff Smith, around 35.6 of : Homotopy Limit Functors on Model Categories and Homotopical Categories, Mathematical Surveys and Monographs 113, AMS (2004) [ISBN: 978-1-4704-1340-8, pdf]

Survey:

• Tim Porter, Section 4.1 of: $S$-Categories, $S$-groupoids, Segal categories and quasicategories [arXiv:math/0401274]

• Clark Barwick, Section 2 of: On (enriched) Bousfield localization of model categories [arXiv:0708.2067]

Further development:

• Clark Barwick, Daniel Kan, A characterization of simplicial localization functors and a discussion of DK equivalences, Indagationes Mathematicae Volume 23, Issues 1–2, March 2012, Pages 69-79 (doi:10.1016/j.indag.2011.10.001)

• Vladimir Hinich, Dwyer-Kan Localization Revisited, Homology, Homotopy and Applications Volume 18 (2016) Number 1 (arXiv:1311.4128, doi:10.4310/HHA.2016.v18.n1.a3)

See also:

• Jacob Lurie, Section A.3.2 of: Higher Topos Theory

• Markus Spitzweck, Homotopy limits of model categories over inverse index categories, Journal of Pure and Applied Algebra Volume 214, Issue 6, June 2010, Pages 769-777 (doi:10.1016/j.jpaa.2009.08.001)

• Aaron Mazel-Gee, Quillen adjunctions induce adjunctions of quasicategories, New York Journal of Mathematics Volume 22 (2016) 57-93 (arXiv:1501.03146, publisher)