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After reaching out to every researcher who cited the manuscript, John Kennison was kind enough to find and scan his copy of the untitled manuscript containing the crude and precise monadicity theorems …

Many papers refer to an untitled manuscript of Jon Beck (Cornell, 1966) for the origin of the monadicity theorem (originally called a "tripleability theorem"). An early proof is in Manes's 1967 thesis …

Dubuc establishes an equivalence between (large) $\mathcal V$-theories and $\mathcal V$-monads on $\mathcal V$, which in particular implies the classical result. … I haven't found an earlier reference with proofs. (1, 2, 3) As far as I am aware, the only paper in which both results follow directly is Lucyshyn-Wright's Enriched algebraic theories and monads for a …

The double category of pseudoalgebras, lax and colax morphisms is defined in §5.4 of Grandis and Paré's Multiple categories of generalised quintets. As far as I'm aware, it is the only reference for t …

Power–Cattani–Winskel's A Representation Result for Free Cocompletions in particular seems promising, but the characterisation result there still assumes that such 2-monads exist in the first place. …

A useful result due to Linton is that for a cocomplete category $C$ and monad $T$ on $C$, if the category of algebras $C^T$ admits reflexive coequalisers, then it is cocomplete (see here for a sketch …

Yes, the Eilenberg–Moore object for a monad $T$ can be presented in terms of two equifiers of the inserter $\mathbf{Ins}(T, 1)$. Denoting by $\phi \colon TU \Rightarrow U$, we equify $1_U$ and $\phi \ …

Is there an intrinsic characterisation of those monads on $\mathbf E$ that are algebraically-free? …

Given a small category $\mathbb C$, we can form the free cocompletion $\mathbf y : \mathbb C \to \mathcal P(\mathbb C)$ and the free completion $\mathbf y^\circ : \mathbb C \to \mathcal P^\circ(\mathb …

Let $T$ be a monad on an accessible category (i.e. an $\mathbf{Ind}$-category). If the underlying endofunctor of $T$ is finitary (i.e. preserves filtered colimits), then the Eilenberg–Moore category o …

There is a full subcategory $\text{IdemMnd}(C) \hookrightarrow \text{Mnd}(C)$ of the category of monads on $C$ spanned by the idempotent monads. …

The category of (finitary) $S$-sorted algebraic theories is equivalent to the category of (finitary) monads on $\mathbf{Set}/S$. … The category of (finitary) $S$-sorted essentially algebraic theories is equivalent to the category of [some class of] monads on [some category]. …

This is a special case of the general construction of cocompletions that preserve existing colimits. The general statement can be found as Theorem 6.23 of Kelly's Basic Concepts of Enriched Category T …

That is, can we always choose to work with 2-monads rather than pseudomonads, so long as we consider their pseudoalgebras, rather than their strict algebras? …

Commutative monads were introduced by Kock in the 1970 paper Monads on symmetric monoidal closed categories. … Though Kock cites Linton's paper, I could find no reference to commutative algebraic theories in any of his papers on commutative monads. …