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*).



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> 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.
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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.



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> 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.
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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.



Characterisation of essentially algebraic theories as monads
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.




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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](https://mathoverflow.net/questions/353258/characterisation-of-essentially-algebraic-theories-as-monads#comment885847_353258).


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).


