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additive functor

Context

Enriched category theory

Could not include enriched category theory - contents

Additive and abelian categories

Homological algebra

homological algebra

and

nonabelian homological algebra

Context

Basic definitions

Stable homotopy theory notions

Constructions

Lemmas

diagram chasing

Homology theories

Theorems

Contents

Definition

Definition

A functor F:π’œβ†’β„¬F: \mathcal{A} \to \mathcal{B} between additive categories is itself called additive if it preserves finite biproducts.

That is,

  1. FF maps a zero object to a zero object, F(0)≃0βˆˆβ„¬F(0) \simeq 0 \in \mathcal{B};

  2. given any two objects x,yβˆˆπ’œx, y \in \mathcal{A}, there is an isomorphism F(xβŠ•y)β‰…F(x)βŠ•F(y)F(x \oplus y) \cong F(x) \oplus F(y), and this respects the inclusion and projection maps of the direct sum:

x y i Xβ†˜ ↙ i y xβŠ•y p x↙ β†˜ p y x y↦FF(x) F(y) i F(x)β†˜ ↙ i F(y) F(xβŠ•y)β‰…F(x)βŠ•F(y) p F(X)↙ β†˜ p F(y) F(x) F(y) \array { x & & & & y \\ & {}_{\mathllap{i_X}}\searrow & & \swarrow_{\mathrlap{i_y}} \\ & & x \oplus y \\ & {}^{\mathllap{p_x}}\swarrow & & \searrow^{\mathrlap{p_y}} \\ x & & & & y } \quad\quad\stackrel{F}{\mapsto}\quad\quad \array { F(x) & & & & F(y) \\ & {}_{\mathllap{i_{F(x)}}}\searrow & & \swarrow_{\mathrlap{i_{F(y)}}} \\ & & F(x \oplus y) \cong F(x) \oplus F(y) \\ & {}^{\mathllap{p_{F(X)}}}\swarrow & & \searrow^{\mathrlap{p_{F(y)}}} \\ F(x) & & & & F(y) }
Remark

In practice, functors between additive categories are generally assumed to be additive.

Examples

Example

The hom-functor Hom(βˆ’,βˆ’):π’œ opΓ—π’œβ†’AbHom(-,-) \colon \mathcal{A}^{op}\times \mathcal{A} \to Ab is additive (and in both arguments separately).

Example

For π’œ=R\mathcal{A} = RMod and Nβˆˆπ’œN \in \mathcal{A}, the functor that forms tensor product of modules (βˆ’)βŠ—N:π’œβ†’π’œ(-)\otimes N \colon \mathcal{A} \to \mathcal{A}.

In fact thes examples are generic, see prop. 2 below.

Properties

Relation to AbAb-enriched functors

An additive category canonically carries the structure of an Ab-enriched category where the AbAb-enrichment structure is induced from the biproducts as described at biproduct.

Proposition

With respect to the canonical Ab-enriched category-structre on additive categories π’œ\mathcal{A}, ℬ\mathcal{B}, additive functors F:π’œβ†’β„¬F : \mathcal{A} \to \mathcal{B} are equivalently Ab-enriched functors.

Proof

An AbAb-enriched functor preserves all finite biproducts that exist, since finite biproducts in Ab-enriched categories are Cauchy colimits.

Characterization of right exact additive functors

Let R,Rβ€²R, R' be rings.

The following is the Eilenberg-Watts theorem. See there for more.

Proposition

If an additive functor F:RF : RMod β†’Rβ€²\to R'Mod is a right exact functor, then there exists an Rβ€²R'-RR-bimodule BB and a natural isomorphism

F≃BβŠ— R(βˆ’) F \simeq B \otimes_R (-)

with the functor that forms the tensor product with BB.

This is (Watts, theorem 1),

References

  • Charles Watts?, Intrinsic characterizations of some additive functors, Proceedings of the American Mathematical Society (1959) (JSTOR)

Revised on March 11, 2014 02:53:28 by Urs Schreiber (89.204.155.115)