Exact category in the sense of Barr

Summary

A category is exact in the sense of Barr, or Barr-exact, when it is a regular category with effective equivalence relations. This means that every internal equivalence relation is the kernel pair of some morphism. So quotients by equivalence relations exist and behave as in sets. Every elementary topos, every abelian category, the category of sets and every variety of algebras such as groups or rings are Barr-exact. This notion is different from the exact categories of Quillen used in algebraic K-theory.