# 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.