Additive functor

Additive functor is a term from the mathematical branch of category theory . These are functors between pre-additive categories that define group homomorphisms between the morphism groups.

definition

Let there be and pre-additive categories. A functor is called additive if the mappings for two objects and are made up of group homomorphisms. ${\displaystyle {\mathfrak {C}}}$ ${\displaystyle {\mathfrak {D}}}$ ${\displaystyle F:{\mathfrak {C}}\rightarrow {\mathfrak {D}}}$ ${\displaystyle \mathrm {Mor} _{\mathfrak {C}}(X,Y)\rightarrow \mathrm {Mor} _{\mathfrak {D}}(FX,FY);\,f\mapsto Ff}$ ${\displaystyle X}$ ${\displaystyle Y}$ ${\displaystyle {\mathfrak {C}}}$

Often one considers additive functors on additive or Abelian categories , since these have further properties on such categories. Most naturally occurring functors between pre-additive categories are additive.

characterization

For functors between Abelian categories to the following characterization has: A functor is if and additive if for all objects from where the equality is to mean: Is a direct sum so well . ${\displaystyle F:{\mathfrak {A}}\rightarrow {\mathfrak {B}}}$ ${\displaystyle F(A_{1}\oplus A_{2})=F(A_{1})\oplus F(A_{2})}$ ${\displaystyle A_{1},A_{2}}$ ${\displaystyle {\mathfrak {A}}}$ ${\displaystyle (\iota _{j}:A_{j}\rightarrow A_{1}\oplus A_{2})_{j=1,2}}$ ${\displaystyle (F\iota _{j}:FA_{j}\rightarrow F(A_{1}\oplus A_{2}))_{j=1,2}}$

Examples

• The Hom functors from the category of - modules over a ring to the category of abelian groups, a fixed module, is additive. The same goes for the functors ${\displaystyle \mathrm {Hom} _{R}(A,-)}$ ${\displaystyle {\mathfrak {M}}_{R}}$ ${\displaystyle R}$ ${\displaystyle R}$ ${\displaystyle {\mathfrak {Ab}}}$ ${\displaystyle A}$ ${\displaystyle R}$ ${\displaystyle \mathrm {Hom} _{R}(-,A):{\mathfrak {M}}_{R}\rightarrow {\mathfrak {Ab}}}$
• The tensor functors are additive, as well ${\displaystyle (A\otimes _{R}-):{\mathfrak {M}}_{R}\rightarrow {\mathfrak {Ab}}}$ ${\displaystyle (-\otimes _{R}A):{\mathfrak {M}}_{R}\rightarrow {\mathfrak {Ab}}}$
• Semi-exact functors are additive.
• The functor with for each module and for each morphism is not additive. ${\displaystyle F:{\mathfrak {M}}_{R}\rightarrow {\mathfrak {M}}_{R}}$ ${\displaystyle FA=A\oplus R}$ ${\displaystyle A}$ ${\displaystyle Ff=f\oplus \mathrm {id} _{R}}$ ${\displaystyle f}$

properties

Additive functors between Abelian categories have the following properties:

• Additive functors convert zero objects into zero objects.
• Additive functors convert finite direct sums into direct sums.
• If you have a short exact sequence and an additive functor, you have a long exact sequence ${\displaystyle 0\rightarrow A\rightarrow A^{'}\rightarrow A^{''}\rightarrow 0}$ ${\displaystyle F}$

${\displaystyle \ldots \rightarrow L_{n}FA\rightarrow L_{n}FA^{'}\rightarrow L_{n}FA^{''}\rightarrow \ldots \rightarrow L_{0}FA\rightarrow L_{0}FA^{'}\rightarrow L_{0}FA^{''}\rightarrow 0}$,

where stand for the -th left derivative . In particular, the 0-th left derivative of an additive functor is right exact . ${\displaystyle L_{n}}$ ${\displaystyle n}$

• Is a sequence of additive functors and natural transformations and and is the sequence for every projective module ${\displaystyle F{\xrightarrow {\rho }}F^{'}{\xrightarrow {\sigma }}F^{''}}$ ${\displaystyle \rho }$ ${\displaystyle \sigma }$ ${\displaystyle P}$

${\displaystyle 0\rightarrow FP{\xrightarrow {\rho ^{P}}}F^{'}P{\xrightarrow {\sigma ^{P}}}F^{''}P\rightarrow 0}$

exactly, you have a long exact sequence for any module ${\displaystyle A}$

${\displaystyle \ldots \rightarrow L_{n}FA\rightarrow L_{n}F^{'}A\rightarrow L_{n}F^{''}A\rightarrow \ldots \rightarrow L_{0}FA\rightarrow L_{0}F^{'}A\rightarrow L_{0}F^{''}A\rightarrow 0}$.

Individual evidence

1. ^ Peter Hilton: Lectures in Homological Algebra. American Mathematical Society, 2005, ISBN 0-8218-3872-5 , set 3.1.
2. ^ Peter Hilton: Lectures in Homological Algebra. American Mathematical Society, 2005, ISBN 0-8218-3872-5 , set 3.2.
3. ↑ Götz Brunner: Homological Algebra. BI-Wissenschaftsverlag, 1973, ISBN 3-411-014420-2 , Chapter III, sentence 23.
4. ↑ Götz Brunner: Homological Algebra. BI-Wissenschaftsverlag, 1973, ISBN 3-411-014420-2 , Chapter III, sentence 24.
5. ^ Peter Hilton: Lectures in Homological Algebra. American Mathematical Society, 2005, ISBN 0-8218-3872-5 , Theorem 3.6.
6. ^ Peter Hilton: Lectures in Homological Algebra. American Mathematical Society, 2005, ISBN 0-8218-3872-5 , Theorem 3.8.