Locally presentable categories over a base, $S$-sorted limit theories, and cartesian first-order theories

The usual Lawvere theories are intrinsically single-sorted but admit both an $S$-sorted generalization and an unsorted analogue (namely small categories with finite products). The situation in the existing literature on locally presentable categories is rather different: $\alpha$-limit theories are small categories with $\alpha$-small limits, seen as unsorted theories, and they correspond under the Gabriel-Ulmer duality to locally $\alpha$-presentable (L$\alpha$P) categories, while a set of sorts $S$ appears only in certain related syntactic theories: Indeed, an L$\alpha$P category $𝒜$ is equivalently the category of models of

(1) an $S$-sorted $\alpha$-ary cartesian theory [1, 2, 3] for some set $S$, and

In this talk, we fix a set $S$ and develop two sharper correspondences that refine the above, namely correspondences between each of the following triples of concepts, after first defining the italicized terms:

1. I.

   (i) $S$-sorted $\alpha$-limit theories, (ii) $S$-sorted L$\alpha$P categories, (iii) $S$-sorted $\alpha$-ary cartesian theories,

2. II.

   (i) e.a. $S$-sorted $\alpha$-limit theories, (ii) e.a. $S$-sorted L$\alpha$P categories, (iii) $S$-sorted $\alpha$-ary e.a. theories,

where we write e.a. as an abbreviation of essentially algebraic. Explicitly, (I.i) and (II.i) are defined as $\alpha$-limit theories equipped with suitable morphisms of $\alpha$-limit theories $\tau :{({\text{Set}}^S)}_{\alpha }^{\text{op}}\to 𝒯$, and (I.ii) and (II.ii) are L$\alpha$P categories $𝒜$ equipped with suitable functors $𝒜\to {\text{Set}}^S$ that are necessarily faithful. More generally, we define special classes of 1-cells of L$\alpha$P categories $𝒜\to 𝒞$ (called single-sorted and e.a. 1-cells) that specialize to (I-II.ii) by taking $𝒞={\text{Set}}^S$ and correspond under the Gabriel-Ulmer duality to special classes of morphisms of $\alpha$-limit theories.

These results have the advantage of providing a more nuanced ‘dictionary’ that relates distinct notions of logical theory to corresponding distinct categorical concepts. Moreover, by regarding categories of models of cartesian theories as concrete categories over ${\text{Set}}^S$, we are able to provide categorical formulations of logical aspects of such categories that involve operations and relations, whose arities are necessarily objects of ${\text{Set}}^S$. We show that every $\alpha$-presentable object $A$ of such a category admits a presentation $A\cong ⟨x\mid \phi ⟩$ in terms a family of epigenerators $x$ and an $\alpha$-ary cartesian formula $\phi$. We use our results to shed light on questions of whether compactness and completeness theorems are available in the setting of $\alpha$-ary cartesian logic (recalling that they are in general unavailable for $\alpha$-ary first-order theories). In turn, we apply these results to prove further results on the question of whether every faithful $1$-cell of L$\alpha$P categories is single-sorted.