See also derived hom space
A category with weak equivalences or homotopical category is a category equipped with the information that some of its morphisms, specifically, a subcategory , are to be regarded as “weakly invertible”. One way to make this notion precise is through the concept of simplicial localization:
The simplicial localization of a category is an (∞,1)-category realized concretely as a simplicially enriched category which is such that the original category injects into it, , such that every morphism in that is labeled as a weak equivalence becomes an actual equivalence in the sense of morphisms in (∞,1)-categories in . And is in some sense universal with this property.
(where gives the 0th simplicial homotopy groupoid).
If the homotopical structure on extends to that of a (combinatorial) model category, then there is another procedure to obtain a simplicially enriched category from , the (∞,1)-category presented by a combinatorial model category. This -category is equivalent to the one obtained by simplicial localization but typically more explicit and more tractable.
Let be the forgetful functor that sends a (small) category to its underlying reflexive graph, and let be its left adjoint. We then get a comonad on , and as usual this defines a functor from Cat to simplicial objects in Cat equipped with a canonical augmentation, where .
The standard resolution of a small category is defined to be the simplicial category .
Note that this is also a simplicial category in the strong sense, i.e. is discrete! Thus we may also regard as an sSet-category. This is a resolution in the sense that the augmentation is a Dwyer-Kan equivalence. (In fact, for objects and in , the morphism admits an extra degeneracy and hence a contracting homotopy.)
The standard simplicial localization of a relative category is the simplicial category where .
This appears as (DwyerKanLocalizations, def. 4.1). Again, is a simplicial category in the strong sense, because is.
Let be a category with weak equivalences. For any two objects, write
for the simplicial set which is the nerve of the category defined as follows:
objects are equivalence classes of zig-zags of morphisms in
where the left-pointing morphisms are to be in ;
morphisms are equivalence classes of “natural transformations” between such objects, fixing the endpoints:
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 three objects, there is an evident compositing morphism
given by horizontally concatenating hammock diagrams as above.
The simplicially enriched category obtained this way is the hammock localization of .
This appears as (DwyerKanCalculating, def. 2.1).
There is an equivalence of categories
This appears as (DwyerKanCalculating, prop. 3.1).
Let be a category with weak equivalences, and let
be a weak equivalence. Then for all objects we have that the to concatenation operations on hammocks induce weak homotopy equivalences
This appears as (DwyerKanCalculating, prop. 3.3).
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.
Then for two homotopical functors (functors respecting the weak equivalences, i.e. ) with
This is (DwyerKanComputations, prop. 3.5).
Let be a full subcategory such that
is homotopy-essentially surjective: for every object there is an object and a weak equivalence ;
Then we have an equivalence of (∞,1)-categories
Let be a simplicial model category. Write for the full -subcategory on the fibrant and cofibrant objects.
Then and are connected by an equivalence of (∞,1)-categories.
This is one of the central statements in (DwyerKanFunctionComplexes). The weak homotopy equivalence between and is in corollary 4.7. The equivalence of -categories stated above follows with this and the diagram of morphisms of simplicial categories in prop. 4.8.
The original articles are
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:
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: