nLab
simplicially enriched category

Context

Enriched category theory

Homotopy theory

(,1)(\infty,1)-Category theory

Contents

Idea

A simplicially enriched category is a category with a simplicial set of morphisms between any two objects.

One may think of the 1-cells in a hom-simplicial set as a 2-morphism, the 2-cells as a 3-morphism and generally a (k1)(k-1)-cell as a k-morphism. Therefore simplicially enriched categories may serves as models for ∞-categories. Precisely which notion of \infty-category depends on which extra structure and property one imposes.

For instance

Definition

Definition

A simplicially enriched category is a category enriched over the cartesian monoidal category sSet of simplicial sets.

Remark

These sSetsSet-enriched categories are sometimes, somewhat imprecisely, called just simplicial categories.

There is a related notion of simplicial groupoid with the added requirement that all arrows in the categories concerned are isomorphisms.

As models for (,1)(\infty,1)-categories

Since simplicial sets that are Kan complexes are an incarnation of ∞-groupoids, an sSetsSet-category all whose hom-objects happen to be Kan complexes may be regarded as a category enriched in ∞-groupoids. By the logic of (n,r)-category theory this should be a model for an (∞,1)-category.

Treating simplicial categories this way as models for (,1)(\infty,1)-categories is one of the central tools in homotopy coherent category theory.

Indeed, there is a model structure on simplicial categories whose fibrant objects are Kan-complex-enriched categories, and which is one model for the (∞,1)-category of (∞,1)-categories.

By a web of Quillen equivalences this is related to the other incarnations of (,1)(\infty,1)-categories. Notably to quasi-categories and complete Segal spaces. For more on this see

Properties

Simplicial localization

To every category with weak equivalences (C,W)(C,W) is associated its simplicial localization L WCL_W C, which is an sSetsSet-category with the property that its homotopy category of an (∞,1)-category coincides with the homotopy category Ho W(C)Ho_W(C).

Model structure

There is a model structure on sSet-categories that presents the (∞,1)-category (∞,1)Cat.

Homotopy Kan extension

The notion of homotopy Kan extension and hence in particular that of homotopy limit and homotopy colimit has a direct formulation in terms of Kan-complex-enriched categories. See homotopy Kan extension for more..

Presentation of (,1)(\infty,1)-topos theory

All of (∞,1)-topos theory can be modeled in terms of sSetsSet-categories. (ToënVezzosi). There is a notion of sSet-site CC that models the notion of (∞,1)-site and a model structure on sSet-enriched presheaves on sSetsSet-sites that is a presentation for the ∞-stack (∞,1)-toposes on CC.

As models for (,2)(\infty,2)-categories

See (∞,2)-category for the moment.

References

The original references on homotopy theory in terms of sSetsSet-categories are

  • William Dwyer, Dan Kan, Simplicial localization of categories, J. Pure and Appl. Algebra 17 (1980), 267-284.

  • William Dwyer, Dan Kan, Equivalences between homotopy theories of diagrams , in Algebraic topology and algebraic K-theory, Annals of Math. Studies 113, Princeton University Press, Princeton, 1987, 180-205.

Simplicially enriched categories as models for (,1)(\infty,1)-categories are discussed in some detail in section A.3 of

as well as in section 2 of

Homotopy coherent category theory on sSetsSet-categories is discussed in

which describes resolutions of the simplicial functor categories between two simplicial categories and

  • Michael Batanin, Homotopy coherent category theory and A A_\infty-structures in monoidal categories (pdf)

which shows that these resolved functor categories are in fact sSetsSet-A-∞ categories.

Revised on February 22, 2012 21:11:56 by Tim Porter (95.147.237.215)