Exact functor
Exact functor is a mathematical term from category theory .
definition
An additive , covariant functor is called
- semi- exact if is exact
- left exact if is exact
- right exact , if is exact
- exact , if is exact
for all short exact sequences in .
A contravariant functor is called half / left / right / exact if it is a covariant functor .
Semi-exact functors between Abelian categories are additive functors .
Examples
- The Hom functors and are left exact.
- The tensor product functors and are right exact .
- The functor “global cuts” on the category of sheaves from Abelian groups into the category of Abelian groups is exact left, see sheaf cohomology .
- For a finite group , the functor “ G -invariants” from the category of the modules into the category of the Abelian groups is exact left, see group cohomology .
- The dual space function in the category of Banach spaces with the continuous linear mappings as morphisms is exact, as can be seen from the theorem of the closed image .
- For any natural number is the functor
- on the category of Abelian groups is additive and contains mono- and epimorphisms, but is not exact.
Individual evidence
- ^ Peter Hilton: Lectures in Homological Algebra. American Mathematical Society, 2005, ISBN 0-8218-3872-5 , definition 3.1.
- ↑ Götz Brunner: Homological Algebra. BI-Wissenschaftsverlag, 1973, ISBN 3-411-014420-2 , Chapter III, Definition 32.
- ^ Peter Hilton: Lectures in Homological Algebra. American Mathematical Society, 2005, ISBN 0-8218-3872-5 , set 3.2.