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It is well known (e.g. Palmgren–Vickers's Partial Horn logic and cartesian categories) that many-sorted essentially algebraic theories (equivalently partial Horn theories / quasi-equational theories / cartesian theories) may be characterised as small finitely complete categories. Here, the set of sorts is not fixed, and varies according to the category.

For a fixed set $S$, is there a known category theoretic characterisation of $S$-sorted essentially algebraic theories, so that (for example) one-sorted essentially algebraic theories may be characterised as certain finitely complete categories?

(Note that I am assuming that the equivalence between quasi-equational/partial Horn/essentially algebraic/cartesian theories continues to hold when the set of sorts is fixed. If this is not the case, I shall update my question accordingly.)

There are some related results in the literature – I've listed the ones I know of below – but I have not found a characterisation for essentially algebraic theories.

  • Keane proves in Abstract Horn theories that a small category is equivalent to a one-sorted Horn theory iff it is finitely complete and has an object $I$ which is $\mathcal M$-injective and such that every object is a $\mathcal M$-subobject of $I^n$ for some $n \in \mathbb N$, for $\mathcal M$ a left-exact class of monomorphisms.
  • Adámek and Porst prove in Algebraic Theories of Quasivarieties that a small category is equivalent to the theory of a one-sorted quasivariety iff it is finitely complete and has an object $I$ which is regular injective and such that every object is a regular subobject of $I^n$ for some $n \in \mathbb N$.
  • $S$-sorted algebraic theories are well-known to be equivalent to strict identity-on-objects finite product-preserving functors from the free category with finite products on $S$.

I should note that a characterisation is deducible from the results of Di Liberti–Loregian–Nester–Sobociński's Functorial Semantics for Partial Theories, as the finitely complete categories induced by (Cauchy complete) $S$-sorted partial algebraic theories; however, this is not a very direct characterisation and I am hoping for one along the lines of the previous examples.

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    $\begingroup$ @TimCampion: $S$-sorted algebraic theory here means a "cartesian category $\mathscr A$ together with a strict identity-on-objects finite product-preserving functor $\mathbb A(S) \to \mathscr A$", where $\mathbb A(S)$ is the free strict cartesian category on the set $A$, as in Bénabou's Structures algébriques dans les catégories. $\endgroup$
    – varkor
    Commented Feb 4, 2021 at 22:12
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    $\begingroup$ @TimCampion: a categorical formulation is not obvious (which is the main motivation for the question), so for the purposes of this question, an $S$-sorted essentially algebraic theory is defined syntactically (i.e. as quadruple $\Gamma = (\Sigma, E, \Sigma_t, \mathrm{Def})$): here, there is an explicit set of sorts. $\endgroup$
    – varkor
    Commented Feb 4, 2021 at 22:27
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    $\begingroup$ Let me note that there is an intuitive characterisation of the categories representing these theories: they are those finitely complete categories $\mathscr C$ with an injective-on-objects functor $F : S \to \mathscr C$, such that every object is the equaliser of a finite product of objects in the image of $F$. This is really the definition I want to capture, but it is not obvious to me how to make this precise in a manner as elegant as the examples in the question. $\endgroup$
    – varkor
    Commented Feb 4, 2021 at 22:29
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    $\begingroup$ In Sec 5 of our paper we say how to pass from an essentially algebraic theory, presented as a partial equational theory to a DCRC, then one can go to DCRC to categories with finite limits taking cc. I am not sure one can do better than this. $\endgroup$
    – Ivan Di Liberti
    Commented Feb 4, 2021 at 22:34
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    $\begingroup$ @varkor. Yes you can. See Sec 10.1 in John Bourke's PhD Thesis: math.muni.cz/~bourkej/papers/JohnBThesis.pdf. $\endgroup$
    – Ivan Di Liberti
    Commented Feb 10, 2021 at 15:23

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