Equivalences of categories and a model structure on relative categories - Lee, Seunghun (Theory and applications of categories, 2025)
We show that there is a model structure on the category RelCat of small relative categories such that for a morphism f in RelCat, f is a weak equivalence iff the associated functor on homotopy 1-categories is an equivalence of categories. In this model category (i) every object is cofibrant and (ii) the homotopy category functor becomes a fibrant replacement. The model structure is left-induced from the model category on small categories with equivalences of categories as weak equivalences by the homotopy category functor in a Quillen equivalent way.