nLab exact category

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.

Definition

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.

Remark

See familial regularity and exactness for a generalization of exactness and its relationship to extensivity.

Properties

Remark

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.

Remark

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 $f\colon B\to A$.

Examples

• 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.

Related concepts

• exact 2-category

• Puppe exact category

• Barr embedding theorem

• regular category

• exact functor

References

• 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