nLab regular coverage

Definition

For a regular category $C$, the regular coverage on $C$ is the coverage in which each covering family has just one element which is a regular epimorphism.

The Grothendieck topology generated from a regular coverage is called the regular topology.

It is the subcanonical Grothendieck topology whose covering families are generated by single regular epimorphisms: the regular coverage.

Properties

• If $C$ is exact or has pullback-stable reflexive coequalizers, then its codomain fibration is a stack for this topology (the necessary and sufficient condition is that any pullback of a kernel pair is again a kernel pair).

• For $\mathcal{C}_{\mathbb{T}}$ the syntactic category of a regular theory, the regular coverage makes it the syntactic site, which is a site of defininition for the classifying topos of $\mathbb{T}$.