nLab simplicial object in Cat

Definition

A simplicial object in $Cat$ is a simplicial object, in to the 1-category Cat of categories and functors between them, equivalently an internal category in sSet.

This would deserve to be and sometimes is called a simplicial category, but more often the latter terminology is used to refer to simplicially enriched categories only (which may be regarded as the special case of simplicial objects in $Cat$ whose simplicial set of objects happens to be simplicially constant).

Related concepts

• simplicial category

  • simplicially enriched category

  • simplicial object in Cat

• simplicial groupoid

• relation between quasi-categories and simplicial categories

References

One of the few references that discusses a model category structure on $Cat^{\Delta^{op}}$ as opposed to just $sSet Cat$ (but see at canonical model structure on Cat) is:

• William Dwyer, Dan Kan, Jeff Smith, Prop. 7.2 in: Homotopy commutative diagrams and their realizations, Journal of Pure and Applied Algebra 57 (1989) 5-24 [doi:10.1016/0022-4049(89)90023-6]

A model category stucture on $Cat(sSet)$ presenting (infinity,1)-categories:

• Geoffroy Horel, A model structure on internal categories, Theory and Applications of Categories, 30 20 (2015) 704-750 [tac:30-20, arXiv:1403.6873]