nLab
exact category

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.

Definition

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.

Remarks

• 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 $f:B\to A$.

Examples

• Any topos is an exact category.

• Any category which is monadic over a power of Set is exact.

• Any abelian category is exact.

• One can construct, for any regular category $C$, a “free” exact category $C_{\mathrm{ex}/\mathrm{reg}}$ on $C$ by adjoining formal quotient objects for congruences. One way to define $C_{\mathrm{ex}/\mathrm{reg}}$ is as the (locally discrete) 2-category whose objects are congruences in $C$ and whose morphisms are anafunctors. If $C$ is already exact, then $C_{\mathrm{ex}/\mathrm{reg}}$ is equivalent to $C$. See regular and exact completions.

• Similarly, one can construct the “free” exact category $C_{\mathrm{ex}/\mathrm{lex}}$ on any category $C$ with finite limits, or even with weak finite limits. The exact categories of the form $C_{\mathrm{ex}/\mathrm{lex}}$ 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

References

• related entries: Barr embedding theorem, regular category
• Carboni, A. and Vitale, E. M. Regular and exact completions, JPAA 125, 1998.
• M. Barr, Exact categories, in: Exact categories and categories of sheaves, Springer Lec. Notes in Math. 236 (1971) 1-120, doi
• Francis Borceux, Dominique Bourn, Mal'cev, protomodular, homological and semi-abelian categories