nLab additive functor

Definition

Definition

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

That is,

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

2. given any two objects $x, y \in \mathcal{A}$, 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:

$$\array { x & & & & y \\ & {}_{\mathllap{i_x}}\searrow & & \swarrow_{\mathrlap{i_y}} \\ & & x \oplus y \\ & {}^{\mathllap{p_x}}\swarrow & & \searrow^{\mathrlap{p_y}} \\ x & & & & y } \quad\quad\stackrel{F}{\mapsto}\quad\quad \array { F(x) & & & & F(y) \\ & {}_{\mathllap{i_{F(x)}}}\searrow & & \swarrow_{\mathrlap{i_{F(y)}}} \\ & & F(x \oplus y) \cong F(x) \oplus F(y) \\ & {}^{\mathllap{p_{F(X)}}}\swarrow & & \searrow^{\mathrlap{p_{F(y)}}} \\ F(x) & & & & F(y) }$$

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 $\mathcal{A} \to \mathcal{B}$ preserves biproducts (where $\mathcal{A}$ and $\mathcal{B}$ are categories with a zero object):

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

Examples

Example

The hom-functor $Hom(-,-) \colon \mathcal{A}^{op}\times \mathcal{A} \to Ab$ is additive in both arguments separately (using the nature of biproducts and that hom-functors preserve limits in each variable separately).

Example

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

In fact these examples are generic, see prop. below.

Example

Every solid abelian group is by definition an additive functor.

Properties

Relation to $Ab$-enriched functors

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

Proposition

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

Proof

An $Ab$-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 $R, R'$ be rings.

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

Proposition

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

$$F \simeq B \otimes_R (-)$$

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

This is (Watts, theorem 1),

Related concepts

• additive monad

• 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, 11 1 (1960) 5-8 (1959) [jstor:2032707]