κ-ary regular and exact categories
arity class: unary, finitary, infinitary
regularity
regular category = unary regular
coherent category = finitary regular
geometric category = infinitary regular
exactness
exact category = unary exact
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 extensive category has coproducts that are disjoint and stable under pullback.
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.
Having filtered colimits which commute with finite limits is also an exactness property.
Exact categories with pullback-stable reflexive coequalizers are an exactness notion.
The various types of additive and abelian categories can also be considered “exactness properties” in a looser sense, though they are not lex colimits in the formal sense. See also AT-category.
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.