nLab
additive functor

Context

Enriched category theory

Could not include enriched category theory - contents

Additive and abelian categories

Homological algebra

homological algebra

(also nonabelian homological algebra)

Introduction

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.

Remark

Each of the following conditions is sufficient for guaranteeing that a functor π’œβ†’β„¬\mathcal{A} \to \mathcal{B} preserves biproducts (where π’œ\mathcal{A} and ℬ\mathcal{B} are categories with a zero object):

  • The functor preserves finite products (for instance, because it’s a right adjoint) and any product in ℬ\mathcal{B} is a biproduct.
  • The functor preserves finite coproducts (for instance, because it’s a left adjoint) and any coproduct in ℬ\mathcal{B} is a biproduct.
  • The functor preserves finite products and coproducts.

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-structure 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 April 27, 2016 14:54:39 by Urs Schreiber (131.220.184.222)