I was wondering if there is a model-theoretic way of defining the infinity category of infinity-groupoid objects in a category $C$ (more generally, if $C$ is an infinity category itself, but, right now a 1-category is enough). Is there a model structure on $C^{\Delta^{op}}$ such that those objects which are fibrant and cofibrant correspond to "internal Kan-complexes" in the correct way? So e.g. I want the C-enriched nerve of an actual groupoid object of C to be fibrant and cofibrant in this model structure. If you don't know the answer in general, for now I am mostly interested in the case that $C$ is the 1-category of topological spaces (here I DO NOT want to think of $C$ as being the same thing as infinity-groupoids or simplicial sets, I actually care about the topology).

More generally, if $C$ is an infinity-category associated to a model category $D$, does this correspond to the Reedy-model structure on $D^{\Delta^{op}}$?

EDIT: I should really be asking for a SIMPLICIAL model category structure.