regular category = unary regular
coherent category = finitary regular
geometric category = infinitary regular
The term exact category has several different meanings. This page is about exact categories in the sense of Barr, also called “Barr-exact categories” or “effective regular categories.” This is distinct from the notion of Quillen exact category.
An exact category (in the sense of Barr) is a regular category in which every congruence is a kernel pair (that is, every internal equivalence relation is effective). Exact categories are also called effective regular categories.
If is a congruence which is the kernel pair of , then if is the image factorization of , one can show that is a coequalizer of . Therefore, congruences have quotients in an exact category. However, not every parallel pair of morphisms need have a coequalizer, and there are also regular categories having all coequalizers which are not exact.
The codomain fibration of an exact category is a stack for its regular topology. However, being exact is not a necessary condition for this to hold in a regular category; all that is required is that if is a kernel pair, then so is for any .
Any topos is an exact category.
One can construct, for any regular category , a “free” exact category on by adjoining formal quotient objects for congruences. One way to define is as the (locally discrete) 2-category whose objects are congruences in and whose morphisms are anafunctors. If is already exact, then is equivalent to . See regular and exact completions.
Similarly, one can construct the “free” exact category on any category with finite limits, or even with weak finite limits. The exact categories of the form for a category with weak finite limits are exactly those which have enough (regular) projectives; in this case the projective objects are the retracts of objects of (Carboni-Vitale 1998). See regular and exact completions.