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The following correspondence between algebraic theories and monads on $\mathbf{Set}$ is well-known (see, for example, Algebraic Theories: A Categorical Introduction to General Algebra).

The category of (finitary) $S$-sorted algebraic theories is equivalent to the category of (finitary) monads on $\mathbf{Set}/S$. The category of models for a fixed algebraic theory $T$ is equivalent to the category of algebras for the corresponding monad.

I would like to know if there is a known analogous result in the setting of essentially algebraic (i.e. finite limit) theories. That is, some statement like the following.

The category of (finitary) $S$-sorted essentially algebraic theories is equivalent to the category of [some class of] monads on [some category]. The category of models for a fixed algebraic theory $T$ is equivalent to the category of algebras for the corresponding monad.

There are many characterisations of categories of models of essentially algebraic theories (i.e. locally presentable categories), but I have not been able to find one in terms of a category of algebras for some monad. Given the many generalisation of Linton's result, for example in Notions of Lawvere theory and Monads with arities and their associated theories, it would seem like this result should be a straightforward application of an existing theorem, but, as far as I can tell, the case of essentially algebraic theories is never explicitly treated. (I originally read Theorem 6.7 of Notions of Lawvere Theory as stating that one-sorted essentially algebraic theories also correspond to finitary monads on $\mathbf{Set}$, but this seems unlikely to be correct.)

If the $S$-sorted case is unknown, I am also interested in the correspondence specifically in the one-sorted setting.

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    $\begingroup$ Is "finitary monad on a presheaf category" a satisfying answer ? An alternative is iterated monads (considering finitary monads acting on category of another finitary monads on sets) $\endgroup$ – Simon Henry Feb 21 at 18:38
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    $\begingroup$ You can have two distinct essentially algebraic theories such that the monad defined by the free-forgetful adjunction to $\mathbf{Set}$ is the same. For example, both abelian groups and torsion-free groups are essentially algebraic theories, but they define the same monad on $\mathbf{Set}$, because every free abelian group is torsion-free. $\endgroup$ – Robert Furber Feb 22 at 1:46
  • $\begingroup$ Of course, I meant to say "torsion-free abelian groups" in my previous comment. $\endgroup$ – Robert Furber Feb 22 at 20:50
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    $\begingroup$ To expand on Simon Henry's comment, the category of models $C$ of an essentially algebraic theory form a locally finitely-presentable theory, which is a finitarily reflective subcategory of a presheaf category $Set^{C_0^{op}}$ (in particular, the inclusion is monadic). In turn, the presheaf category $Set^{C_0^{op}}$ is monadic, via the obvious forgetful functor, over $Set^{Ob C_0}$. Thus $C$ admits a functor to a slice of $Set$ which is a composite of monadic functors (but typically not itself monadic). I'm not sure how to get a composite of monadic functors to $Set$ itself though... $\endgroup$ – Tim Campion Feb 22 at 22:02
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I'm going to give a partial answer to my question, which addresses a misconception I had and illustrates why many of the existing generalisations of theory–monad correspondence are not sufficient to provide a monadic correspondence with essentially-algebraic theories. As far as I know, this indicates that a correspondence is open.


Let $\mathbb C$ be a small category, and denote by $\widehat{\mathbb C}$, $\mathbf{Rex}(\mathbb C)$, $\mathbf{Ind}(\mathbb C)$ and $\mathbf{Lex}(\mathbb C)$ the free cocompletion, finite cocompletion, filtered cocompletion and finite completion of $\mathbb C$, respectively. We have $\mathbf{Ind}(\mathbf{Rex}(\mathbb C)) \simeq \widehat{\mathbb C}$.

We'll instantiate the results of Bourke & Garner's Monads and theories for $K : \mathbf{Rex}(\mathbb C) \hookrightarrow \widehat{\mathbb C}$. This is an example of the so-called "presheaf context". The results follow directly, so I won't spell everything out. $K$ preserves finite colimits and so a $\mathbf{Rex}(\mathbb C)$-theory is a finite colimit-preserving identity-on-objects functor $J : \mathbf{Rex}(\mathbb C) \to \mathcal T$. A model is a finite-limit preserving functor $F : \mathcal T^{\mathrm{op}} \to \mathbf{Set}$. A $\mathbf{Rex}(\mathbb C)$-nervous monad is a monad on $\mathbf{Rex}(\mathbb C)$ whose underlying endofunctor preserves filtered colimits. The categories of $\mathbf{Rex}(\mathbb C)$-nervous monads and $\mathbf{Rex}(\mathbb C)$-theories are equivalent by Theorem 17 of ibid. (and similarly their categories of algebras and categories of models by Theorem 34 of ibid.)

We can now take opposites appropriately to generalise the classical algebraic theory–finitary monad correspondence: the categories of finite limit-preserving identity-on-objects functors from $\mathbf{Lex}(\mathbb C^{\mathrm{op}})$ (which I'll dub "category-sorted lex theories") are equivalent to finitary monads on $\widehat{\mathbb C}$. When $\mathbb C = S$ is discrete, category-sorted lex theories are equivalent to sorted algebraic theories (this can be seen as a consequence of sifted colimits and filtered colimits coinciding on indexed sets, or of the coincidence of finite completion and finite product completion on sets). When $\mathbb C$ is a non-discrete category, the two constructions differ. This essentially gives a classification of finitary and sifted colimit-preserving monads on presheaf categories (at least on small categories), which was suggested by Simon Henry.

I was previously assuming that category-sorted lex theories (at least for discrete categories) was the right notion of essentially algebraic theory. However, I was mistaken: category-sorted lex theories are less expressive than sorted algebraic theories. In particular, the requirement for theories to be identity-on-objects is too restrictive, as the codomains may no longer be finitely complete. An analogous definition of sorted essentially algebraic theory to sorted algebraic theory will look different (as far as I'm aware, no such characterisation has been given in the literature) and, as such, the existing theory–monad correspondences are insufficient to describe a correspondence for essentially algebraic theories. I'll continue to pursue this question according to this line of thinking, but unless anyone can see that I've missed something obvious, I'm satisfied that such a correspondence is at least not known (or easily derivable from results in the literature).

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