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For the purposes of this question, let us assume that the categories involved are locally presentable, and the monads are accessible (i.e. preserve $\kappa$-filtered colimits for some cardinal $\kappa$ … Now suppose we have a lax morphism of monads $(F, \varphi) : (\mathbf C, S) \to (\mathbf D, T)$, comprising a functor $F \colon \mathbf C \to \mathbf D$ and a natural transformation $\varphi : TF \Rightarrow …

I believe the original reference for this fact is Theorem 2 of Maranda's 1966 On Fundamental Constructions and Adjoint Functors, although the terminology is not modern. A more readable reference is Th …

It's not clear to me precisely what you are asking, but in response to the question in the title, there are now two proposed type theories for virtual double categories. Virtual equipment type theory …

Let $T$ be a 2-monad on a nice 2-category $\mathcal K$, so that the inclusion $T\text{-}\mathbf{Alg}_s \to T\text{-}\mathbf{Alg}_l$ of the 2-category of (strict) $T$-algebras and strict $T$-algebra mo …

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 …

A Frobenius monad is a monad in $\mathbf{DagCat}$ satisfying an additional law (see §5 of Heunen–Karvonen's Monads on dagger categories). …

For every pseudomonad $T$ on the 2-category of (locally small) categories $\mathbf{Cat}$, we can consider the 2-category of $T$-pseudoalgebras and pseudomorphisms $T\text{-PsAlg}_p$, which is equipped …

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 \ …

These are known as pseudo-distributive laws and are the most common notion of distributive law of 2-dimensional monads, even when both 2-dimensional monads in question are strict (i.e. 2-monads rather …

I have reworded Proposition 11 ibid. below in terms of finitary monads rather than algebraic theories. Proposition (Faro–Kelly). … The proof appears to generalise at least to arbitrary commutative monads with rank on $\mathbf{Set}$. …

Yes, Theorem 3.4 of Gambino–Lobbia's On the formal theory of pseudomonads and pseudodistributive laws establishes that pseudomonad morphisms are in correspondence with liftings to pseudoalgebras. They …

Typically, limits of multisorted algebraic theories (by which I mean a pair of a set $S$ and an $S$-sorted algebraic theory $\mathbb F(S) \to L$) are most easily described in terms of their presentati …

Therefore, a functor between categories of algebras for two monads (regardless of rank) corresponds to a monad morphism if and only if the functor commutes with the forgetful functors. … In fact, the corresponding statement is true for monads on any category, not just $\mathrm{Set}$. …

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. …

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 …