Simplicial Enrichment of Joyal-Tierney Model Structure on $Cat$

Question

The Joyal Tierney model structure on $Cat$ (sometimes called the ``canonical" model structure) is (as mentioned in https://sbseminar.wordpress.com/2012/11/16/the-canonical-model-structure-on-cat/) proper, cartesian, simplicial, combinatorial, and every object is both fibrant and cofibrant. Is there a reference for the simplicial enrichment of $Cat$?

Answer 1

I asked Chris Schommer-Pries (the author of the blog post) about this. It's discussed in an unpublished note by Charles Rezk (either scroll down here to the note "A model category for categories", or else use this direct link. Only available in dvi, it seems).

It's important to note that the simplicial enrichment on the homset $\mathsf{Cat}(C,D)$ comes by taking the nerve of the groupoid of functors from $C$ to $D$ and natural isomorphisms -- not the category of functors from $C$ to $D$ and all natural transformations. If you took the nerve of the full functor category, you would presumably get something which is "a simplicially enriched model category with respect to the Joyal model structure on $\mathsf{sSet}$", but not a simplicially enriched model category in the usual sense, i.e. with respect to the Quillen model structure on $\mathsf{sSet}$.