regular category = unary regular
coherent category = finitary regular
geometric category = infinitary regular
An exactness property of a category asserts the existence of certain limits and colimits, and moreover that the limits and colimits interact in a certain way. Frequently, this includes stability of the colimits under pullback, and also a condition expressing that some of the input data can be recovered from the colimit.
Many types of exactness can be expressed in terms of “colimits in the left-exact world”.
Exactness properties of a functor refer to preservation of limits or colimits of certain kind, existence of adjoints and possibly their exactness properties.
As particular cases of familial regularity and exactness we have:
An adhesive category has pushouts of monomorphisms that are stable under pullback and “van Kampen”.
An exhaustive category has colimits of sequences of monomorphisms that are pullback-stable and “exhaust” the colimit.
More generally, having colimits of some class which are van Kampen is an exactness property.
Exact categories with pullback-stable reflexive coequalizers are an exactness notion.
A site can be considered as a category with “exactness structure”, or as a way of specifying certain exactness conditions which ought to hold after “completion”. See postulated colimit? and exact completion.