The term exact category has several different meanings. This page is about exact categories in the sense of Barr 1971, also called “Barr-exact categories” or “effective regular categories.” This is distinct from the notion of Quillen exact category.
∞-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 exact category (in the sense of Barr 1971) is a regular category in which every congruence is a kernel pair (that is, every internal equivalence relation is effective). These are also called effective regular categories.
See familial regularity and exactness for a generalization of exactness and its relationship to extensivity.
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.
The codomain fibration of an exact category is a stack (2-sheaf) 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.
More generally, so is every pretopos, by definition.
Any regular category with van Kampen quotients of congruences is exact. See at van Kampen colimit for a proof.
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.
If $C$ is exact and $T$ is a Lawvere theory, then the category $Mod(T, C)$ of $T$-models in $C$ is also exact. See Theorem 5.11 of Barr’s Exact Categories.
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.
Michael Barr, Pierre Grillet, Donovan van Osdol, Exact Categories and Categories of Sheaves, Lec. Notes in Math. 236, Springer (1971) [doi:10.1007/BFb0058579]
Michael Barr, Exact categories, in: Exact categories and categories of sheaves, Springer Lec. Notes in Math. 236 (1971) 1-120 [doi:10.1007/BFb0058580, pdf, pdf]
Aurelio Carboni, Enrico Vitale, Regular and exact completions, JPAA 125, 1998.
Francis Borceux, Dominique Bourn, Mal'cev, protomodular, homological and semi-abelian categories
Last revised on January 25, 2024 at 01:20:58. See the history of this page for a list of all contributions to it.