Exact functor

Exact functor is a mathematical concept from category theory.

Definition

An additive, covariant functor is called

  • Semi- exact if is exactly
  • Left exact if is exactly
  • Quite precisely, if is exactly
  • Precisely, if is exactly

For all short exact sequences.

A contravariant functor is called semi / left / right / exactly, if it this is a covariant functor.

Half Exact functors between abelian categories are additive functors.

Examples

  • The Hom - functors and are left exact.
  • The tensor product functors and are quite accurate.
  • The functor " global sections" on the category of sheaves of abelian groups to the category of abelian groups is left exact, see Garbenkohomologie.
  • For a finite group of the functor "G- invariant " from the category of left -modules is exact in the category of abelian groups, see Gruppenkohomologie.
  • The dual space functor in the category of Banach spaces with the continuous linear maps as morphisms is exact, as is clear from the set of the completed image.
  • For an arbitrary natural number is the functor
Abelian category Hom functor Group cohomology Closed range theorem
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