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This is one of the main themes of Arkor–Bourke–Ko's paper Enhanced 2-categorical structures, two-dimensional limit sketches and the symmetry of internalisation. In particular, in §5, the authors intro …

Let $f : S \to T$ be a morphism of algebraic theories. Such a morphism induces a monadic functor $f^* : \mathrm{Mod}(T) \to \mathrm{Mod}(S)$ (hence $f^*$ has a left adjoint). We may view $f$ syntactic …

The answer to your question really depends on just how closely you want the theory of "algebraic theories relative to $\mathcal M$" to mirror the theory of classical ($S$-sorted) algebraic theories. P …

The original reference for the essential algebraicity of elementary toposes is Freyd's Aspects of topoi (in which the notion of essentially algebraic theory, which is equivalent to that of a finite li …

These claims are proven more generally for the category $\mathrm{Mnd}_f(\mathscr A)$ of finitary monads on a locally presentable category $\mathscr A$ in Lack's On the monadicity of finitary monads. ( …

Commutative algebraic theories were introduced by Linton in the 1966 paper Autonomous Equational Categories. Commutative monads were introduced by Kock in the 1970 paper Monads on symmetric monoidal c …

Algebraic theories (by which I mean the formalism based on bijective-on-objects functors) and abstract clones both capture universal algebraic structure, and are well-known to be equivalent. Algebraic …

(1, 2, 3) Though Linton's An outline of functorial semantics does contain the essence of the results and proofs of the monad–theory correspondence (see in particular Theorems 8.1 and 9.1 – 9.3), it is …

I am looking for a copy of Michel Thiébaud's 1971 thesis Self-Dual Structure-Semantics and Algebraic Categories, which appears to be an early reference for the relationship between the Kleisli constru …

There is a duality between locally strongly finitely presentable (LSFP) categories and (Cauchy complete) cartesian categories, i.e. multisorted algebraic theories [1]. The internal logic of cartesian …

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 / …

Let $T : \mathbf{Set}^\mathrm{op} \to \mathscr T$ be a large equational theory (i.e. a bijective-on-objects product-preserving functor). Following Linton in Some Aspects of Equational Categories, we c …

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 p …

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 catego …