equivalences in/of $(\infty,1)$-categories
See also derived hom space
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 $C$:
(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.
See simplicial localization of a homotopical category.
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 $G_n C = G^{n+1} C$.
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.)
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.
Let $(C,W)$ be a category with weak equivalences. For $X,Y \in C$ any two objects, write
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 $C$
where the left-pointing morphisms are to be in $W$;
its morphisms are “natural transformations” between such objects, fixing the endpoints:
$L^H C(X,Y)$ 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
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).
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 $\pi_0 L^H C(X,Y)$.
There is an equivalence of categories
This appears as (DwyerKanCalculating, prop. 3.1).
Let $(C,W)$ be a category with weak equivalences, and let
be a weak equivalence. Then for all objects $U \in C$ we have that the to concatenation operations on hammocks induce weak homotopy equivalences
and
This appears as (DwyerKanCalculating, prop. 3.3).
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.
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
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
This is (DwyerKanComputations, prop. 3.5).
Let $i : (C_1, W_1) \hookrightarrow (C_2, W_2)$ be a full subcategory such that
$i$ 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 $c_2 \stackrel{\simeq}{\to} i(c_1)$;
there is a functor $Q : (C_2,W_2) \to (C_1, W_1)$ and a natural transformation
Then we have an equivalence of (∞,1)-categories
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.
Let $C$ be a simplicial model category. Write $C^\circ$ for the full $Set$-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 $\infty$-categories stated above follows with this and the diagram of morphisms of simplicial categories in prop. 4.8.
The original articles are
William Dwyer, Daniel Kan, Simplicial localizations of categories , J. Pure Appl. Algebra 17 (1980), 267–284. (pdf)
William Dwyer, Daniel Kan, Calculating simplicial localizations , J. Pure Appl. Algebra 18 (1980), 17–35. (pdf)
William Dwyer, Daniel Kan, Function complexes in homotopical algebra , Topology 19 (1980), 427–440.
William Dwyer, Daniel Kan, 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:
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.