nLab
exact functor

Context

Category theory

Limits and colimits

Homological algebra

homological algebra

and

nonabelian homological algebra

Context

Basic definitions

Stable homotopy theory notions

Constructions

Lemmas

diagram chasing

Homology theories

Theorems

Contents

Idea

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

The term originates in homological algebra, see remark 2 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 MM and short exact sequences 0ABC00 \to A \to B \to C \to 0 of modules (over some ring RR), the sequence

0Mod R(M,A)Mod R(M,B)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()=Mod R(M,)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

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:

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

  • F:CDF: C \to D preserves terminal objects if F(t C)F(t_C) is terminal in DD whenever t Ct_C is terminal in CC;

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

    F(c)F(π 1)F(c×d)F(π 2)F(d)F(c) \stackrel{F(\pi_1)}{\leftarrow} F(c \times d) \stackrel{F(\pi_2)}{\to} F(d)

    exhibits F(c×d)F(c \times d) as a product of F(c)F(c) and F(d)F(d), where π 1:c×dc\pi_1: c \times d \to c and π 2:c×dd\pi_2: c \times d \to d are the product projections in CC;

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

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

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

Terminology

Some author use the term “left exact” when CC 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 equivalent formulated in terms of the behaviour of the functor on short exact sequences. We now discuss this case.

Proposition

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

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

Corollary

In particular for 0ABC00 \to A \to B \to C \to 0 is an exact sequence in the abelian category CC, we have that

  • if FF is left exact then

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

    is an exact sequence in DD;

  • if FF is right exact then

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

    is an exact sequence in DD;

  • if FF is exact then

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

    is an exact sequence in DD.

Also: if FF 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 0AiBpC00 \to A \stackrel{i}{\to} B \stackrel{p}{\to} C \to 0 being an exact sequence is equivalent to ii being a monomorphism and pp being an epimorphism, hence to 0A0 \to A being the kernel of ii, ii being the kernel of pp and C0C \to 0 being the cokernel of pp. Since the functor FF is assumed to preserve this kernel-property, but not the cokernel property, it follows that F(0)F(A)F(0) \to F(A) is the kernel of F(A)F(i)F(B)F(A) \stackrel{F(i)}{\to} F(B), but not more than that. This means that

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

is an exact sequence, as claimed.

Remark

The properties of corollary 1 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.

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 in section 3.3 of

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)

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

Revised on December 27, 2013 10:45:32 by Urs Schreiber (82.113.106.190)