For $C$ a coherent category$C$ the coherent coverage on $C$ is the coverage in which the covering families are generated by finite, jointly regular-epimorphic families. Similarly, on an infinitary-coherent category (a.k.a. a “geometric category”), the infinitary-coherent coverage (a.k.a. geometric coverage) is generated by all small jointly regular-epimorphic families.

The corresponding Grothendieck topology is called the coherent topology, making $C$ into a coherent site.

Equivalently, this coverage is generated by single regular epimorphisms and by finite unions of subobjects (resp. small unions in the infinitary case).

Topoi of sheaves for (finitary) coherent topologies on coherent categories are called coherent toposes. (The terminology is slightly confusing, though, because every topos is a coherent category.) Note that every topos is equivalent to a topos of sheaves for the infinitary coherent topology on an infinitary-coherent site, namely itself.

If $C$ is a pretopos, then its self-indexing is a stack for its coherent topology. Exactness and extensivity are stronger than necessary, however; a pair of necessary and sufficient conditions for this to hold are that

If $R\;\rightrightarrows\; A$ is a kernel pair, then for any $f\colon B\to A$, the pullback $f^*R \;\rightrightarrows\; B$ is also a kernel pair (this is equivalent to the codomain fibration being a stack for the regular topology).

If $A,B \rightarrowtail C$ are a pair of subobjects, then for any $f\colon X\to A$ and $g\colon Y\to B$ and any isomorphism $f^*(A\cap B) \cong g^*(A\cap B)$, the pushout