Canonical model structure on Cat

The canonical model structure on Cat is defined as follows:

• The fibrations are Isofibrations
• The cofibrations are injective-on-objects functors
• The weak equivalences are Equivalences of categories

The factorization of a functor $F:\mathcal{C}\to\mathcal{D}$ into a trivial cofibration and a fibration factors through the essential image of $F$. This is clearly an isofibration, and the map $\mathcal{C}\to\mathrm{Im}(F)$ is an equivalence of categories: we can send $c:\mathcal{C}$ to $(F(c),\mathrm{id})$. Note that is is an equivalence, not an iso!

Conversely, the factorization of a functor $F:\mathcal{C}\to\mathcal{D}$ into a cofibration and a trivial fibration factors through the Cograph.

References

• https://ncatlab.org/nlab/show/canonical+model+structure+on+Cat