The Clowder Project

Let $F\colon \mathcal{C}\to \mathcal{D}$ be a functor.

1. 1.

   Characterisations. The following conditions are equivalent:¹

   1. (a)

      The functor $F$ is a epimorphism of categories.

   2. (b)

      For each morphism $f\colon A\to B$ of $\mathcal{D}$, we have a diagram

      
      in $\mathcal{D}$ satisfying the following conditions:
      1. (i)

         We have $f=\alpha _{0}\circ \phi _{1}$.

      2. (ii)

         We have $f=\psi _{m}\circ \alpha _{2m}$.

      3. (iii)

         For each $0\leq i\leq 2m$, we have $\alpha _{i}\in \operatorname {\mathrm{Mor}}(\mathrm{Im}(F))$.

2. 2.

   Surjectivity on Objects. If $F$ is an epimorphism of categories, then $F$ is surjective on objects.