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.



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 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.


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 RAR\rightrightarrows A is a kernel pair, then so is f *RBf^*R \rightrightarrows B for any f:BAf\colon B\to A.



Last revised on January 25, 2024 at 01:20:58. See the history of this page for a list of all contributions to it.