category of small categories

This is the category of small categories and functors between them. It is the prototype of a 2-category, but here we only treat it as a 1-category.

Properties from the database

Deduced properties

Properties from the database

Deduced properties*

*This also uses the deduced satisfied properties.

There is 1 property for which the database doesn't have an answer if it is satisfied or not. Please help to contribute the data!

isomorphisms: functors that are bijective on objects and morphisms
monomorphisms: faithful functors that are injective on objects
epimorphisms: A functor $F : \mathcal{C} \to \mathcal{D}$ is an epimorphism iff $F$ is surjective on objects and for every morphism $s$ in $\mathcal{D}$ there is a zigzag over $U := F(\mathcal{C})$ , meaning morphisms $u_1,\dotsc,u_{m+1} \in U$ , $v_1,\dotsc,v_m \in U$ , $x_1,\dotsc,x_m \in \mathcal{D}$ and $y_1,\dotsc,y_m \in \mathcal{D}$ such that $s = x_1 u_1$ , $u_1 = v_1 y_1$ , $x_{i-1} v_{i-1} = x_i u_i$ , $u_i y_{i-1} = v_i y_i$ , $x_m v_m = u_{m+1}$ and $u_{m+1} y_m = s$ .
regular monomorphisms:
regular epimorphisms: