nLab simplicial category

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.

The analogous issue arises for “simplicial groupoids” etc., see also there.

Of course there are close relations between these three meanings. In particular simplicially enriched categories may be identified with those simplicial objects in Cat whose face and degeneracy maps are constant on objects and only affect the morphisms.

An example of a text that speaks about what elsewhere are mostly regarded as simplicially enriched categories as simplicial objects in Cat which are constant on objects is (Lydakis 98, around def. 3.2, remark 3.6).

Related concepts

• simplicial category

  • simplicially enriched category

  • simplicial object in Cat

• simplicial groupoid

• relation between quasi-categories and simplicial categories

References

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

• Lydakis, Simplicial functors and stable homotopy theory Preprint, available via Hopf archive, 1998 (pdf)