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).
simplicial object in Cat
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:
A model category stucture on $Cat(sSet)$ presenting (infinity,1)-categories:
Last revised on December 13, 2023 at 18:11:26. See the history of this page for a list of all contributions to it.