nLab
simplicial object in Cat

Definition

A simplicial object in $\mathrm{Cat}$ is a particular case of simplicial object, of course, with the $C$ there being $\mathrm{Cat}$. It therefore has a simplicial set of objects and a simplicial set of morphisms.

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 ${\mathrm{Cat}}^{\Delta }$ as opposed to just $\mathrm{sSet}\mathrm{Cat}$ is

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