Small finitary abelian categories for Hall multiplication

A small finitary abelian category for Hall multiplication is a small abelian category $\mathcal{A}$ (its object and morphism collections are sets, by Small, locally small, and large categories), with abelian structure as in Abelian category, in which the following sets are finite for every pair of objects $M,N$:

1. ${\mathrm{Hom}}_{\mathcal{A}}(M,N)$;
2. ${\mathrm{YExt}}_{\mathcal{A}}^1(M,N)$, the set of Yoneda classes of short exact sequences $0\to N\to E\to M\to 0$. Equivalence is the one-fold instance of the generated endpoint-fixed chain-map relation in Equivalence of n-fold extensions: chain maps are the identity on $N$ and $M$, and the relation is closed under reflexivity, symmetry and transitivity. A short exact sequence and its endpoint order use Exact sequence and short exact sequence in an abelian category, An extension of an object by an object in an abelian category and An n-fold Yoneda extension.

One may instead supply a small skeleton ${\mathcal{A}}_0$ of an essentially small abelian category and perform the construction on ${\mathcal{A}}_0$. Here “essentially small” means that such a small equivalent category is part of the given data; the construction does not select representatives by an unstated choice principle. The isomorphism classes $\mathrm{Iso}({\mathcal{A}}_0)$ then form a set.

No ${\mathbb{F}}_q$-linear structure, enough projectives or injectives, finite number of subobjects of every object, or identification with derived ${\mathrm{Ext}}^1$ is included in this definition. Subobjects are the monomorphism classes of Subobject and quotient object as mutual-factorisation classes of monomorphisms and epimorphisms, and quotients are the cokernels specified in The quotient of an object by a subobject.