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:
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. 4 (using prop. 2) 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: