nLab
simplicial object in Cat

Definition

A simplicial object in $Cat$ is a particular case of simplicial object, of course, with the $C$ there being $Cat$. It therefore has a simplicial set of objects and a simplicial set of morphisms. This is equivalent to giving an internal category in simplicial sets.

Remark A simplicial object in $\Cat$ is sometimes called a simplicial category, but that has other meanings.

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 $Cat^\Delta$ as opposed to just $sSet Cat$ is

• William Dwyer, Dan Kan, Jeff Smith, Homotopy commutative diagrams and their realizations, Journal of Pure and Applied Algebra 57 (1989) 5-24

A model category sturcture on $Cat(sSet)$ presenting (infinity,1)-categories is in

• Geoffroy Horel, A model structure on internal categories (arXiv:1403.6873).