nLab
exact functor

Idea

A functor is left exact or flat if it preserves finite limits, and right exact if it preserves finite colimits.

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 categories with finite limits preserves finite limits if and only if:

• it preserves terminal objects, binary products, and equalizers; or

• it preserves terminal objects and binary pullbacks.

Since these conditions frequently come up individually, it may be worthwhile listing them separately:

• $F:C\to D$ preserves terminal objects if $F(t_C)$ is terminal in $D$ whenever $t_C$ is terminal in $C$;

• $F:C\to D$ preserves binary products if the pair of maps

  $$F(c)\stackrel{F({\pi }_1)}{\leftarrow }F(c\times d)\stackrel{F({\pi }_2)}{\to }F(d)$$F(c) \stackrel{F(\pi_1)}{\leftarrow} F(c \times d) \stackrel{F(\pi_2)}{\to} F(d)

  exhibits $F(c\times d)$ as a product of $F(c)$ and $F(d)$, where ${\pi }_1:c\times d\to c$ and ${\pi }_2:c\times d\to d$ are the product projections in $C$;

• $F:C\to D$ preserves equalizers if the map

  $$F(i):F(e)\to F(c)$$F(i): F(e) \to F(c)

  is the equalizer of $F(f),F(g):F(c)\overrightarrow{\to }F(d)$, whenever $i:e\to c$ is the equalizer of $f,g:c\overrightarrow{\to }d$ in $C$.

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 exact’ is sometimes abbreviated lex. In particular, Lex is the 2-category of categories with finite limits and lex functors. See also continuous functor. Similarly, but more rarely, ‘right exact’ is sometimes abbreviated as rex.

In higher category theory

The notion of exact functor has straightforward analogs in higher category theory.

For (∞,1)-category theory see exact (∞,1)-functor.

References

for instance section 3.3 of

• Kashiwara, Schapira, Categories and Sheaves