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.

Contents

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 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.

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 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.

References

Revised on October 19, 2014 12:53:13 by Thomas Holder (89.204.130.106)