related by the Dold-Kan correspondence
Every category with weak equivalences presents under Dwyer-Kan simplicial localization a simplicially enriched category or alternatively under Charles Rezk’s simplicial nerve a Segal space, both of which are incarnations of a corresponding (∞,1)-category with the same objects of , at least the 1-morphisms of and such that every weak equivalence in becomes a true equivalence (homotopy equivalence) in .
For the purposes of the present entry, we understand under a category with weak equivalences the absolute minimum structure that may deserve to go by that name, namely a relative category:
A morphism of relative catgeories is a functor that preserves weak equivalences.
Write for the category of relative categories and such morphisms between them.
The compatibility of the various nerve and simplicial localization functors is in section 1.11 of
Relative categories; another model for the homotopy theory of homotopy theories – Part I: the model structure (pdf)
Relative categories; another model for the homotopy theory of homotopy theories – Part II: the weak equivalences (pdf)
Partial model categories and their simplicial nerves, a relative Yoneda embedding, and a Quillen theorem for homotopy pullbacks (pdf)