nLab exact functor

This entry is about the concept of “exact functors” in plain category theory. For the different concept of that name in triangulated category theory see at triangulated functor.

Idea

A left/right exact functor is a functor that preserves finite limits/finite colimits.

The term originates in homological algebra, see remark below, where a central role is played by exact sequences (originally of modules, more generally in any abelian category) 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

$$0 \to Mod_R(M, A) \to Mod_R(M,B) \to Mod_R(M,C)$$

is exact – but note that there is no 0 on the right hand. Thus $F(-) = Mod_R(M,-)$ 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. Below we give the general definition and then discuss the relation to the concept in homological algebra in the section Properties - On abelian categories.

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.

Properties

Characterisations

General

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 $(d/F)^{op}$ 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

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

  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

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

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

Between categories of modules

Right exact functors between categories of modules are characterized by the Eilenberg-Watts theorem. See there for more details.

On abelian categories / in homological algebra

In the context of homological algebra, the notion of left/right exact functors is considered specifically in abelian categories. In this context the above formulation is equivalently formulated in terms of the behaviour of the functor on short exact sequences. We now discuss this case.

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

• if $F$ is left exact then

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

  is an exact sequence in $D$;

• if $F$ is right exact then

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

  is an exact sequence in $D$;

• if $F$ is exact then

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

  is an exact sequence in $D$.

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

Proof

We discuss the first case. The second is formally dual. The third combines the two cases.

For the first case notice that $0 \to A \stackrel{i}{\to} B \stackrel{p}{\to} C \to 0$ being an exact sequence is equivalent to $i$ being a monomorphism and $p$ being an epimorphism, hence to $0 \to A$ being the kernel of $i$, $i$ being the kernel of $p$ and $C \to 0$ being the cokernel of $p$. Since the functor $F$ is assumed to preserve this kernel-property, but not the cokernel property, it follows that $F(0) \to F(A)$ is the kernel of $F(A) \stackrel{F(i)}{\to} F(B)$, but not more than that. This means that

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

is an exact sequence, as claimed.

Remark

The properties of corollary explain the “left”/“right”-terminology: a left exact functor preserves exactness of sequences to the left of a morphism (only), while a right exact functor preserves exactness to the right.

Related concepts

• LexCat

• derived functor

• exact (∞,1)-functor

• satellite

• exact category

References

Early use of left exact and exact:

• Henri Cartan, Samuel Eilenberg, Homological Algebra, Princeton Univ. Press (1956), Princeton Mathematical Series 19 (1999) [ISBN:9780691049915, doi:10.1515/9781400883844, pdf]

• 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.

General discussion

• Francis Borceux, Section 6.1 in: Handbook of Categorical Algebra Vol. 1: Basic Category Theory [doi:10.1017/CBO9780511525858]

• Masaki Kashiwara, Pierre Schapira, Section 3.3 in: Categories and Sheaves

See also:

• Michael Barr, Right exact functors, J. Pure Appl. Algebra 4 (1974) 1–8 [doi:10.1016/0022-4049(74)90025-5, pdf, pdf]

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

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

Discussion of left exactness (or flat functor) in the context of (∞,1)-category theory is in

• Jacob Lurie, def. 5.3.2.1 in Higher Topos Theory