# nLab simplicial category

### Context

#### Enriched category theory

enriched category theory

## Extra stuff, structure, property

### Homotopical enrichment

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

(∞,1)-category theory

# Contents

## Definition

The term simplicial category has at least three common meanings. Thus, to avoid ambiguity, it is perhaps better to avoid it entirely and use an equivalent, unambiguous term for the particular meaning one has in mind.

First of all

• The simplicial category $\Delta$ is the domain category for the presheaf category of simplicial sets. This is the full subcategory of Cat on categories which are linear quivers, or equivalently the category of finite, non-empty totally ordered sets and order-preserving functions between them. To avoid ambiguity, $\Delta$ may also be called the simplex category or the simplicial indexing category.

More importantly, there are these two uses of the word:

1. A simplicial category is a simplicial object in Cat (that is, a functor from $\Delta^{op}$ to $Cat$), just like a simplicial set is a simplicial object in Set, a simplicial space is a simplicial object in Top, a simplicial abelian group is a simplicial object in Ab, and so on. To avoid ambiguity, simplicial objects in Cat may be called exactly that. They may equivalently be regarded as internal categories in simplicial sets.

2. A simplicial category also frequently means a category enriched over the category of simplicial sets (Quillen 67, II.1), i.e. an sSet-enriched category. Such categories can be identified with simplicial objects in Cat all of whose face and degeneracy morphisms are bijective on objects. To avoid ambiguity, categories enriched over simplicial sets may be called simplicially enriched categories.