regular category = unary regular
coherent category = finitary regular
geometric category = infinitary regular
More generally, for a regular cardinal, a -geometric category, or -coherent category, is a regular category with unions for -small families of subobjects, stable under pullback. (For this reduces to the notion of coherent category, called a pre-logos by Freyd–Scedrov.)
See familial regularity and exactness for a general description of the spectrum from regular categories through finitary and infinitary coherent categories.
Frequently, geometric categories are additionally required to be well-powered. If a geometric category is well-powered, then its subobject posets are complete lattices, hence also have all intersections. Moreover, by the adjoint functor theorem for posets, it is a Heyting category.
However, since geometric logic does not include implication or infinite conjunction, this categorical structure should not necessarily be expected to exist in a category called “geometric” (and when they do exist, they are not preserved by geometric functors). A requirement of well-poweredness is also inconsistent with the spectrum of familial regularity and exactness.
Note, however, that if a geometric category has a small generating set, then it is necessarily well-powered. In particular, this applies to the syntactic category of any (small) geometric theory, and also to any Grothendieck topos.
Around lemma A1.4.18 in