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 and short exact sequences of modules (over some ring ), the sequence
is exact – but note that there is no 0 on the right hand. Thus converts an exact sequence into a left exact sequence; such a functor is called a left exact functor. Dually, one has right exact functors.
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.
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.
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.
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:
preserves terminal objects if is terminal in whenever is terminal in ;
preserves binary products if the pair of maps
exhibits as a product of and , where and are the product projections in ;
preserves equalizers if the map
is the equalizer of , whenever is the equalizer of in .
‘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.
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.
In particular for an exact sequence in the abelian category , we have that
if is left exact then
is an exact sequence in ;
if is right exact then
is an exact sequence in ;
if is exact then
is an exact sequence in .
Also: if is exact then it preserves chain homology.
We discuss the first case. The second is formally dual. The third combines the two cases.
For the first case notice that being an exact sequence is equivalent to being a monomorphism and being an epimorphism, hence to being the kernel of , being the kernel of and being the cokernel of . Since the functor is assumed to preserve this kernel-property, but not the cokernel property, it follows that is the kernel of , but not more than that. This means that
is an exact sequence, as claimed.
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.
An early use of left exact and exact is in:
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