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.



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 RX×XR\hookrightarrow X\times X is a congruence which is the kernel pair of f:XYf:X \to Y, then if f=mpf = m \circ p is the image factorization of ff, one can show that pp is a coequalizer of RR. 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 RAR\rightrightarrows A is a kernel pair, then so is f *RBf^*R \rightrightarrows B for any f:BAf\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.

  • If CC is exact and TT is a Lawvere theory, then the category Mod(T,C)Mod(T, C) of TT-models in CC 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 CC, a “free” exact category C ex/regC_{ex/reg} on CC by adjoining formal quotient objects for congruences. One way to define C ex/regC_{ex/reg} is as the (locally discrete) 2-category whose objects are congruences in CC and whose morphisms are anafunctors. If CC is already exact, then C ex/regC_{ex/reg} is equivalent to CC. See regular and exact completions.

  • Similarly, one can construct the “free” exact category C ex/lexC_{ex/lex} on any category CC with finite limits, or even with weak finite limits. The exact categories of the form C ex/lexC_{ex/lex} for a category CC with weak finite limits are exactly those which have enough (regular) projectives; in this case the projective objects are the retracts of objects of CC (Carboni-Vitale 1998). See regular and exact completions.


  • 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 (pdf)

Revised on February 21, 2017 12:28:53 by Todd Trimble (