Finite-field hereditary Hall categories

A finite-field hereditary Hall category is a supplied small ${\mathbb{F}}_q$-linear abelian category $\mathcal{A}$ with the following properties:

1. For every pair $M,N$, ${\mathrm{Hom}}_{\mathcal{A}}(M,N)$ is a finite-dimensional ${\mathbb{F}}_q$-vector space and the Yoneda extension group ${\mathrm{YExt}}^1(M,N)$ is finite-dimensional over ${\mathbb{F}}_q$. Its addition is Baer sum; scalar multiplication by $\lambda \in {\mathbb{F}}_q$ is pushout along $\lambda {\mathrm{id}}_N$.
2. For every $M,N$, ${\mathrm{YExt}}^2(M,N)=0$, where a two-fold extension is an exact sequence $0\to N\to E_1\to E_0\to M\to 0$ modulo endpoint-fixed chain-map equivalence (An n-fold Yoneda extension, Equivalence of n-fold extensions).
3. Smallness and the finiteness conditions of the Hall definition hold (Small finitary abelian categories for Hall multiplication).

Here ${\mathbb{F}}_q$-linear means that every Hom set is a vector space and composition is bilinear. “Hereditary” in this page means exactly the stated vanishing of Yoneda ${\mathrm{Ext}}^2$; no projective or injective resolutions, derived-Ext identification, or choice principle is built into the definition. Under these conditions the Euler form used below is $⟨[M],[N]⟩={\dim }_{{\mathbb{F}}_q}\mathrm{Hom}(M,N)-{\dim }_{{\mathbb{F}}_q}{\mathrm{YExt}}^1(M,N)\mathrm{.}$