nLab
simplicial localization

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Context

Homotopy theory

homotopy

homotopy theory

paths and cylinders

(∞,1)-Category theory

(∞,1)-category theory

Background

Basic concepts

Universal constructions

Local presentation

Theorems

Extra stuff, structure, properties

Models

See also derived hom space

Simplicial localisation

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⊃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 LC 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↪LC, 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 LC. And LC is in some sense universal with this property.

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

Ho C(a,b)≃Π 0(LC(a,b))Ho_C(a,b) \simeq \Pi_0 (L C(a,b))

(where Π 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 (∞,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.

Hammock localization

… (see section 4.1 in the reference below) …

Properties

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.

Proof This is 2.5 in

  • Dwyer, Dan 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 .

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.

References

A comprehensive survey of the general topic involved here can be found in the following paper:

  • Tim Porter, S-Categories, S-groupoids, Segal categories and quasicategories (arXiv)

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:

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

Revised on September 6, 2010 09:26:48 by Urs Schreiber (134.100.32.213)
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