nLab
additive functor

Definition

Definition

A functor $F:𝒜\to ℬ$ between additive categories is itself called additive if it preserves finite biproducts.

That is,

1. $F$ maps a zero object to a zero object, $F(0)\simeq 0\in ℬ$;

2. given any two objects $x,y\in 𝒜$, there is an isomorphism $F(x\oplus y)\cong F(x)\oplus F(y)$, and this respects the inclusion and projection maps of the direct sum:

$$\begin{array}{ccccc}x& & & & y\\ & {}_{i_X}\searrow & & {\swarrow }_{i_y}\\ & & x\oplus y\\ & {}^{p_x}\swarrow & & {\searrow }^{p_y}\\ x& & & & y\end{array}\stackrel{F}{\mapsto }\begin{array}{ccccc}F(x)& & & & F(y)\\ & {}_{i_{F(x)}}\searrow & & {\swarrow }_{i_{F(y)}}\\ & & F(x\oplus y)\cong F(x)\oplus F(y)\\ & {}^{p_{F(X)}}\swarrow & & {\searrow }^{p_{F(y)}}\\ F(x)& & & & F(y)\end{array}$$\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 $\mathrm{Hom}(-,-):{𝒜}^{\mathrm{op}}\times 𝒜\to \mathrm{Ab}$ is additive (and in both arguments separately).

Example

For $𝒜=R$Mod and $N\in 𝒜$, the functor that forms tensor product of modules $(-)\otimes N:𝒜\to 𝒜$.

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

Properties

Relation to $\mathrm{Ab}$-enriched functors

An additive category canonically carries the structure of an Ab-enriched category where the $\mathrm{Ab}$-enrichment structure is induced from the biproducts as described at biproduct.

Proposition

With respect to the canonical Ab-enriched category-structre on additive categories $𝒜$, $ℬ$, additive functors $F:𝒜\to ℬ$ are equivalently Ab-enriched functors.

Proof

An $\mathrm{Ab}$-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 $R,R\prime$ be rings.

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

Proposition

If an additive functor $F:R$Mod $\to R\prime$Mod is a right exact functor, then there exists an $R\prime$-$R$-bimodule $B$ and a natural isomorphism

$$F\simeq B{\otimes }_R(-)$$F \simeq B \otimes_R (-)

with the functor that forms the tensor product with $B$.

This is (Watts, theorem 1),

Related concepts

• enriched functor

• In the context of derived functors in homological algebra one considers functors that are additive and in addition left/right exact functors, as discussed above in Characterization by exactness.

• satellite

References

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