Exact functor

A functor that commutes with finite limits and colimits. More precisely, an additive functor $ F : \mathfrak A \rightarrow \mathfrak B $ between Abelian categories $ \mathfrak A $ and $ \mathfrak B $ is called exact if it maps a short exact sequence

$$ 0 \rightarrow A \rightarrow B \rightarrow C \rightarrow 0 $$

in $ \mathfrak A $ into a short exact sequence

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

in $ \mathfrak B $.

If $ \mathfrak A $ and $ \mathfrak B $ are non-Abelian categories, then a functor $ F : \mathfrak A \rightarrow \mathfrak B $ is sometimes called exact if it maps a commutative diagram

$$ A \begin{array}{c} \rightarrow ^ { {\epsilon _ 1} } \\ \mathop \rightarrow \limits _ { {\epsilon _ {2} }} \end{array} B \mathop \rightarrow \limits ^ \nu C $$

in $ \mathfrak A $, where $ ( \epsilon _ {1} , \epsilon _ {2} ) $ is the kernel pair of $ \nu $, and $ \nu $ is the cokernel of the pair $ ( \epsilon _ {1} , \epsilon _ {2} ) $, into a diagram in $ \mathfrak B $ with the same properties.

Comments

In the general theory of categories, a functor is commonly called left exact if it preserves (i.e. commutes with) all finite limits, right exact if it preserves all finite colimits, and exact if it is both left and right exact. An additive functor between Abelian categories automatically preserves finite products and coproducts; so the question of exactness for such a functor reduces to that of the preservation of kernels and cokernels, or equivalently of exact sequences — whence the name. For functors between non-Abelian categories, there are several conflicting uses of the term "exact" , including the one given in the final sentence of the main article above; but the one given in the first sentence of this addendum is the most widely understood.

In Russian literature there is some confusion between the terms "exact functor" and "faithful functor" , cf. also Faithful functor and the references given there.