TITLE: When do the (Set valued) models of a Lawvere Theory form an abelian category?
QUESTION [6 upvotes]: By a "Lawvere Theory", I mean a category $\mathbb{T}$ with finite products and a distinguished family of objects $(S_\alpha)$, called sorts, where every object of $\mathbb{T}$ is isomorphic to a finite product of some $S_\alpha$s.
Then ($\mathcal{C}$-valued) models of $\mathbb{T}$ are exactly finite-product preserving functors $M : \mathbb{T} \to \mathcal{C}$. Moreover, $\mathbb{T}$-models assemble into a category with natural transformations as the arrows.
I'm curious if there are combinatorial conditions on the category $\mathbb{T}$ which force the category of $\mathsf{Set}$ valued $\mathbb{T}$-models to be abelian. I've made some partial progress (for instance, I think I know when the category of $\mathbb{T}$-models is enriched in $\mathsf{Ab}$), but I can't help but feel like I've been beaten to the punch by several decades. I suspect it will be fairly difficult to get a sufficient condition for the full structure of an abelian category, but I'm also interested in smaller results. For instance, when does the category of $\mathbb{T}$-models have biproducts?
(NB: the biproduct question might be easy -- I haven't thought about it at all yet. I'm just giving it as an example of the kinds of results I would be interested in)

Does anybody know of any work that's been done in this area? I've checked my usual books, as well as the nlab, but I haven't found anything.

Thanks in advance ^_^

REPLY [3 votes]: Abelian categories are characterized by being enriched in abelian groups, having finite limits, and being exact. A reference for this result is M. Barr, Exact categories, Springer Lect. Notes in Math. 2366 (1971) 1-120 (according to Borceux and Bourn's book Mal'cev, Protomodular, Homological and Semi-Abelian Categories). Since categories of models in a finitely complete/exact category are finitely complete/exact, your question reduces to necessary and sufficient conditons for the category of models to be enriched in abelian groups.
If $T$ is locally small, then it is necessary that it itself be enriched in abelian groups. Indeed, in that case $T^{op}$ is isomorphic to the category of representable functors as a full subcategory of all functors to $\mathbf{Set}$. Since representable functors are product-preserving, we see that $T^{op}$ is by by hypothesis a full subcategory of an abelian category, so $T^{op}$ has an enrichment in abeliean groups. But being enriched in abelian groups is evidently a self-dual notion, so $T$ also has an enrichment in abelian groups.
Now because $T$ has finite products, the structure of $T$ being enriched in abelian groups reduces to each object having the structure of an abelian group and every morphism being a homomorphism for this structure. On the one hand, $+\colon X\times X\to X$ is given by $\pi_1+\pi_2\colon X\times X\to X$ where $\pi_i\colon X\times X\to X$ are projection morphisms. The converse holds by, say, the Yoneda lemma.
We thus have

a natural transformation $0\colon\mathbf 1\Rightarrow\mathrm{id}_T$, where $\mathbf 1\colon T\to T$ sends every object to the terminal object, and
a natural transformation $+\colon\mathrm{id}_T\times\mathrm{id}_T\Rightarrow\mathrm{id}_T$

that satisfy the axioms of commutative monoid and have for each $X$ the pair of $+_X\colon X\times X\to X$ and one of the projections $\pi_i\colon X\times X\to X$ be a product structure (this makes the enrichment be in abelian groups rather than just commutative monoids).
Applying a product preserving functor $F$ then results in precisely the structure of an abelian group object on $F$: $F(\mathbf 1)\Rightarrow F$ and $F(+)\colon F\times F\Rightarrow F$. That natural transformations between product-preserving functors preserve this structure is immediate by the structure maps being images of morphisms. Consequently, we have our desired enrichment in abelian groups.