nLab abelianization of an additive category

(under construction)

Idea

The category of small additive presheaves of abelian groups on an additive category contains a subcategory of finitely presented (that is compact) objects. This subcategory has a nice universal property.

Definition

Freyd’s definition

Let $C$ be an additive subcategory. An additive presheaf is finitely presented if it is a cokernel of a morphism of representables. The full subcategory of object parts of these cokernels is $A(C)$, the Freyd abelianization of $C$ (Freyd 1966).

Thus, an object $x$ of $A(C)$ is a presheaf such that there exist an exact sequence of presheaves of abelian groups of the form

$$Hom(-,a) \longrightarrow Hom(-,b) \longrightarrow x \longrightarrow 0 \mathrlap{\,,}$$

where $a,b$ are objects in $C$. In the case where $C$ is abelian already, finitely presentable additive presheaves of abelian groups were studied under the name of coherent functors by Auslander. They are automatically coherent objects in the functor category, namely each finitely generated subobject of a finitely presentable functor is finitely presentable as well. Moreover, for $C$ small abelian, coherent functors are also projective.

Verdier’s definition

Verdier 1967 has reformulated $A(C)$ without a recourse to presheaves. He considers the (additive) arrow category $Arr(C)$ of $C$ and inside it, the morphisms which he calls negligible.

A morphism

$$(g \colon X \to Y,\, h \colon X'\to Y') \colon (f \colon X \to X') \longrightarrow (g \colon Y \to Y')$$

is negligible if the compositions $h f = f' g = 0$ are null morphisms. For any pair $(f,f')$ of arrows, the negligible morphisms $f\to f'$ form a subgroup.

Now, the objects of $A(C)$ are the objects of the arrow category, while the morphisms are the morphisms among the arrows modulo the negligible morphisms (that is the elements of the quotient group). Therefore, $C$ embeds in $Arr(C)$ as $X\mapsto id_X$, which projects $Hom$-wise to $A(C)$. For any fixed additive category $E$, the functor $C\to A(C)$ induces a functor

$$Additive\big(A(C),E\big) \longrightarrow Additive(C,E) \mathrlap{\,.}$$

Literature

• Freyd 1966

• Verdier 1967

• Neeman

The category of coherent functors over an abelian category was studied in

• M. Auslander, Coherent functors, In: Eilenberg, S., Harrison, D.K., MacLane, S., Röhrl, H. (eds) Proceedings of the Conference on Categorical Algebra, La Jolla 1965, Springer, 1966.

Finitely presented additive functors on abelian categories with values in abelian groups are the topic of Ch. 10 in

• Mike Prest, Purity, spectra and localization, Enc. Math. Appl. 112 (2009)