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
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 $R\hookrightarrow X\times X$ is a congruence which is the kernel pair of $f:X \to Y$, then if $f = m \circ p$ is the image factorization of $f$, one can show that $p$ is a coequalizer of $R$. 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.
See familial regularity and exactness for a generalization of exactness and its relationship to extensivity.
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 $R\rightrightarrows A$ is a kernel pair, then so is $f^*R \rightrightarrows B$ for any $f\colon B\to A$.
Any topos is an exact category.
Any category which is monadic over a power of Set is exact. A proof may be found here.
Any abelian category is exact. In fact an abelian category is precisely an exact additive category.
Any slice or co-slice of an exact category is also exact. (Source: Borceux and Bourn, Appendix A.)
One can construct, for any regular category $C$, a “free” exact category $C_{ex/reg}$ on $C$ by adjoining formal quotient objects for congruences. One way to define $C_{ex/reg}$ is as the (locally discrete) 2-category whose objects are congruences in $C$ and whose morphisms are anafunctors. If $C$ is already exact, then $C_{ex/reg}$ is equivalent to $C$. See regular and exact completions.
Similarly, one can construct the “free” exact category $C_{ex/lex}$ on any category $C$ with finite limits, or even with weak finite limits. The exact categories of the form $C_{ex/lex}$ for a category $C$ with weak finite limits are exactly those which have enough (regular) projectives; in this case the projective objects are the retracts of objects of $C$ (Carboni-Vitale 1998). See regular and exact completions.