Exact functor

Exact functor is a mathematical term from category theory .

definition

An additive , covariant functor is called ${\displaystyle F:{\mathfrak {C}}\rightarrow {\mathfrak {D}}}$

• semi- exact if is exact ${\displaystyle FA\rightarrow FA'\rightarrow FA''}$
• left exact if is exact ${\displaystyle 0\rightarrow FA\rightarrow FA'\rightarrow FA''}$
• right exact , if is exact ${\displaystyle FA\rightarrow FA'\rightarrow FA''\rightarrow 0}$
• exact , if is exact ${\displaystyle 0\rightarrow FA\rightarrow FA'\rightarrow FA''\rightarrow 0}$

for all short exact sequences in . ${\displaystyle 0\rightarrow A\rightarrow A'\rightarrow A''\rightarrow 0}$ ${\displaystyle {\mathfrak {C}}}$

A contravariant functor is called half / left / right / exact if it is a covariant functor . ${\displaystyle F:{\mathfrak {C}}\rightarrow {\mathfrak {D}}}$ ${\displaystyle {\mathfrak {C}}^{op}\rightarrow {\mathfrak {D}}}$

Semi-exact functors between Abelian categories are additive functors .

Examples

• The Hom functors and are left exact. ${\displaystyle \mathrm {Hom} (A,-)}$ ${\displaystyle \mathrm {Hom} (-,B)}$
• The tensor product functors and are right exact . ${\displaystyle (A\otimes -)}$ ${\displaystyle (-\otimes B)}$
• 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 . ${\displaystyle G}$ ${\displaystyle G}$
• 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 ${\displaystyle n>1}$

${\displaystyle {\mathfrak {Ab}}\to {\mathfrak {Ab}},\quad M\mapsto nM}$

on the category of Abelian groups is additive and contains mono- and epimorphisms, but is not exact.

Individual evidence

1. ^ Peter Hilton: Lectures in Homological Algebra. American Mathematical Society, 2005, ISBN 0-8218-3872-5 , definition 3.1.
2. ↑ Götz Brunner: Homological Algebra. BI-Wissenschaftsverlag, 1973, ISBN 3-411-014420-2 , Chapter III, Definition 32.
3. ^ Peter Hilton: Lectures in Homological Algebra. American Mathematical Society, 2005, ISBN 0-8218-3872-5 , set 3.2.