Exact functor on abelian categories

Let be abelian categories and be an -functor. Then #m/thm/homology

is left exact iff it preserves kernels; equivalently for any exact sequence the sequence is exact.
is right exact iff it preserves cokernels; equivalently for any exact sequence the sequence is exact.

Thus is exact iff it preserves short exact sequences.

Proof

It suffices to show the left exact case, whence the right exact case follows by duality.

Suppose is left exact and is an exact sequence. Then the designated arrows and are the kernels of and respectively. It follows and are the kernels of and respectively, so the sequence is exact.

For the converse, if for any exact we have exact, then preserves kernels. It must also preserve biproducts since it is -enriched. Thus, by the limit construction theorems we have a left exact functor.

#state/tidy | #lang/en | #SemBr