nLab
exact functor

Idea

A functor is left exact or flat if it preserves finite limits.

Definition

• A functor $F:C\to D$ is right exact if for all objects $d\in D$ the comma category $F/d$ is filtered.

• A functor $F:C\to D$ is left exact if for all objects $d\in D$ the opposite comma category $(d/F{)}^{\mathrm{op}}$ is filtered.

• A functor is exact if it is both left and right exact.

Proposition

• A left exact functor preserves every finite limit that exists in $C$.

• If $C$ admits all finite limits then a functor is left exact if and only if it preserves these limits.

Proposition

• A functor between abelian categories is left exact if and only if it preserves direct sums and kernels.

• A functor between abelian categories is right exact if and only if it preserves direct sums and cokernels.

Terminology

Frequently the term “left exact” is restricted to the case that $C$ has all finite limits. If so, then the general case is called a flat functor.

Conceivably, it might be used also in the more general case, but to refer to a weaker notion: a functor that preserves those finite limits that exist. Certainly that's how I would interpret ‘finitely continuous functor’. —Toby

Left exactness is sometimes abbreviated lex. In particular, $\mathrm{Lex}$ is the 2-category of categories with finite limits and lex functors. See also continuous functor.

References

for instance section 3.3 of

• Kashiwara, Schapira, Categories and Sheaves