Exact Functor

An exact functor is a functor that preserves short exact sequences. In the usual setting of Abelian categories, an additive functor is exact iff it sends every short exact sequence to a short exact sequence (Stacks Project Authors 2026a). The terminology also applies to categories of groups, using their usual short exact sequences.

For an additive functor between Abelian categories, left exactness means that every short exact sequence gives an exact sequence

while right exactness means that it gives an exact sequence

For an additive contravariant functor, left exactness instead means that is an exact sequence, and right exactness means that is an exact sequence. In either variance, a functor is exact iff it is both left and right exact.

For a module over a commutative unit ring , the tensor product functor is right exact, and it is exact iff is a flat module (Stacks Project Authors 2026b). It need not be left exact. For example, applying to the injection gives the zero map (Knapp 2016, p. 587).

For every left unital R-module over a unit ring , the covariant functor and the contravariant functor , with values in Abelian groups, are left exact (Knapp 2016, pp. 585-586). The first is exact iff is projective and the second iff is injective (Stacks Project Authors 2026cd).

More generally, an additive functor between exact categories is exact when it sends each chosen short exact sequence in the source to a chosen short exact sequence in the target. Thus exactness depends on the specified exact structures (Bühler 2010, §5).