Exact Functor - Definitions

Definitions

Let P and Q be abelian categories, and let F: P→Q be an additive functor (so that, in particular, F(0)=0). Let

0→A→B→C→0

be a short exact sequence of objects in P.

If F is a covariant functor, we say that F is

  • half-exact if F(A)→F(B)→F(C) is exact. This is similar to the notion of a topological half-exact functor.
  • left-exact if 0→F(A)→F(B)→F(C) is exact.
  • right-exact if F(A)→F(B)→F(C)→0 is exact.
  • exact if 0→F(A)→F(B)→F(C)→0 is exact.

If G is a contravariant functor from P to Q, we can make a similar set of definitions. We say that G is

  • half-exact if G(C)→G(B)→G(A) is exact.
  • left-exact if 0→G(C)→G(B)→G(A) is exact.
  • right-exact if G(C)→G(B)→G(A)→0 is exact.
  • exact if 0→G(C)→G(B)→G(A)→0 is exact.

It is not always necessary to start with an entire short exact sequence 0→A→B→C→0 to have some exactness preserved; it is only necessary that part of the sequence is exact. The following statements are equivalent to the definitions above:

  • F is left-exact if 0→A→B→C exact implies 0→F(A)→F(B)→F(C) exact.
  • F is right-exact if A→B→C→0 exact implies F(A)→F(B)→F(C)→0 exact.
  • F is exact if A→B→C exact implies F(A)→F(B)→F(C) exact.
  • G is left-exact if A→B→C→0 exact implies 0→G(C)→G(B)→G(A) exact.
  • G is right-exact if 0→A→B→C exact implies G(C)→G(B)→G(A)→0 exact.
  • G is exact if A→B→C exact implies G(C)→G(B)→G(A) exact.

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