Exact (effective) Categories
The theory of equivalence relations is a regular theory. An equivalence relation on an object of a regular category is a monomorphism into that satisfies the interpretations of the conditions for reflexivity, symmetry and transitivity.
Every kernel pair defines an equivalence relation . Conversely, an equivalence relation is said to be effective if it arises as a kernel pair. An equivalence relation is effective if and only if it has a coequalizer and it is the kernel pair of this.
A regular category is said to be exact, or exact in the sense of Barr, or effective regular, if every equivalence relation is effective.
Read more about this topic: Regular Category
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