David Roberts class of admissible maps

Admissible maps are analogues of surjective functions of sets, used to define ‘essential surjectivity’ in a general ambient category for internal functors

Definition

Let $S$ be a category with a singleton pretopology $J$ (i.e. a site). A class of maps $E$ is called admissible for $J$ if it satisfies the following properties

1. $E$ is a singleton pretopology in which $J$ is cofinal
2. $E$ contains all the split epimorphisms in $S$
3. If the composite $x \to y \to z$ is in $E$ and $x\to y$ is a split epimorphism, then $y\to z$ is in $E$.

Example The prototype is the class of $J$-epimorphisms for a pretopology $J$ on a category with pullbacks. This class is admissible for $J$.

Example If $E$ is a saturated singleton pretopology, then it is admissible for itself.

If we drop reference to $J$ in the above definition, $E$ is called a class of admissible maps.