# nLab simplicially enriched category

### Context

#### Enriched category theory

enriched category theory

homotopy theory

## Theorems

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

(∞,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 $(k-1)$-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 $sSet$-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 $(\infty,1)$-categories

Since simplicial sets that are Kan complexes are an incarnation of ∞-groupoids, an $sSet$-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 $(\infty,1)$-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 $(\infty,1)$-categories. Notably to quasi-categories and complete Segal spaces. For more on this see

### Properties

#### Simplicial localization

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

#### 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 $(\infty,1)$-topos theory

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

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

See (∞,2)-category.

## References

In the context of model category theory, simplicially enriched categories (simplicial model categories) appear in

• Dan Quillen, chapter II, section 1 of Homotopical algebra, Lecture Notes in Mathematics 43, Springer-Verlag 1967, iv+156 pp.

The original references on homotopy theory in terms of $sSet$-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 $(\infty,1)$-categories are discussed in some detail in section A.3 of

as well as in section 2 of

Homotopy coherent category theory on $sSet$-categories is discussed in

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

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

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

Revised on June 9, 2017 03:45:11 by Peter Heinig (84.183.88.168)