nLab
exact functor

Idea

In homological algebra, a central role is played by exact sequences (originally of modules) and the fact that various functors preserve or destroy exactness of sequences to some extent gave vital information on those functors.

In this context, one says that an exact functor is one that preserves exact sequences. However, many functors are only “exact on one side or the other”. For instance, for all modules $M$ and short exact sequences $0\to A\to B\to C\to 0$ of modules (over some ring $R$), the sequence

$$0\to {\mathrm{Mod}}_R(M,A)\to {\mathrm{Mod}}_R(M,B)\to {\mathrm{Mod}}_R(M,C)$$0 \to Mod_R(M, A) \to Mod_R(M,B) \to Mod_R(M,C)

is exact – but note that there is no 0 on the right hand. Thus $F(-)={\mathrm{Mod}}_R(M,-)$ converts an exact sequence into a left exact sequence; such a functor is called a left exact functor. Dually, one has right exact functors.

It is easy to see that an additive functor between additive categories is left exact in this sense if and only if it preserves finite limits. Since merely preserving left exact sequences does not require a functor to be additive, in a non-additive context one defines a left exact functor to be one which preserves finite limits, and dually.

Definition

A functor between finitely complete categories is called left exact (or flat) if it preserves finite limits. Dually, a functor between finitely cocomplete categories is called right exact if it preserves finite colimits. A functor is called exact if it is both left and right exact.

Specifically, Ab-enriched functors between abelian categories are exact if they preserve exact sequences.

Characterisations

Proposition

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

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

In other language, this says that a functor between finitely complete categories is left exact if and only if it is (representably) flat. Conversely, one can show that a representably flat functor preserves all finite limits that exist in its domain.

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 $C$.

Terminology

Some author use the term “left exact” when $C$ does not have all finite limits, defining it to mean a flat functor.

‘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.

Left exact functors correspond to pro-representable functors, provided some smallness conditions are satisfied.

Properties

On abelian categories

Proposition

A functor $F:C\to D$ between abelian categories is left exact if and only if it preserves direct sums and kernels.

A functor $F:C\to D$ between abelian categories is right exact if and only if it preserves direct sums and cokernels.

Corollary

In particular for $0\to A\to B\to C\to 0$ is an exact sequence in the abelian category $C$, we have that

• if $F$ is left exact then

  $$0\to F(A)\to F(B)\to F(C)$$0 \to F(A) \to F(B) \to F(C)

  is an exact sequence in $D$;

• if $F$ is right exact then

  $$F(A)\to F(B)\to F(C)\to 0$$F(A) \to F(B) \to F(C) \to 0

  is an exact sequence in $D$;

• if $F$ is exact then

  $$0\to F(A)\to F(B)\to F(C)\to 0$$0 \to F(A) \to F(B) \to F(C) \to 0

  is an exact sequence in $D$.

Also: if $F$ is exact then it preserves chain homology.

Related concepts

• exact functor

• exact (∞,1)-functor

References

An early use of left exact and exact is in:

• A. Grothendieck, 1959, Technique de descente et théorèmes d’existence en géométrie algèbrique. II. Le théorème d’existence en théorie formelle des modules, in Séminaire Bourbaki, Vol. 5 , Exp. No. 195, 369 – 390, Soc. Math. France Numdam, Paris.

A general discussion is for instance section 3.3 of

• Kashiwara, Schapira, Categories and Sheaves

A detailed discussion of how the property of a functor being exact is related to the property of it preserving homology in generalized situations is in

• Michael Barr, Preserving homology , Theory and Applications of Categories, Vol. 16, 2006, No. 7, pp 132-143. (TAC)