n-category = (n,n)-category
n-groupoid = (n,0)-category
A simplicial object in is a particular case of simplicial object, of course, with the there being . It therefore has a simplicial set of objects and a simplicial set of morphisms.
Remark A simplicial object in is sometimes called a simplicial category, but that has other meanings.
simplicial object in Cat
One of the few references that discusses a model category structure on as opposed to just is