Proposition The folk model structure on \mathsf {Cat} [efr-ZPPL]

There is a unique model structure on \mathsf {Cat} where the weak equivalences are the equivalences. It has the following characterization:

1. The cofibrations are those functors which are injective on objects
2. The fibrations are the isofibrations
3. The acyclic fibrations are those Grothendieck fibrations where each fiber is a contractible groupoid
4. Every category is fibrant and cofibrant