# Has anybody studied strict/pseudo morphisms of monads?

There is a notion of morphism from a monad $T:\mathscr C\to \mathscr C$ to another one $T':\mathscr C'\to \mathscr C'$. It arose here on MO e. g. in "Functors between monads": what are these really called? or Distributive law between Kleisli triples. Such morphisms have been considered by Pumplün, Street and probably others, they are functors $F:\mathscr C\to\mathscr C'$ together with a transformation $T'F\to FT$ with some coherence conditions generalizing those for a distributive law. The idea is to have an induced functor between Eilenberg-Moore algebras.

While this is certainly "the" correct notion, I've recently encountered a situation when it is not. With several colleagues we are studying varieties like meet-semilattices-with-a-nucleus; "correct" morphisms $(M,j)\to(M',j')$ must be $f:M\to M'$ which preserve meets and satisfy $j'f=fj$, not just $j'f\leqslant fj$ as it would be if one would view $j$ and $j'$ as monads and take monad morphisms in the above sense.

The natural "categorified" version of such morphisms would then be natural isomorphisms $T'F\cong FT$; that is, a $\textit{pseudo}$ version, while Pumplün-Street notion would then be the $\textit{lax}$ version.

Question: can one distinguish such "strict"/"pseudo" morphisms of monads among more general ones by some abstract-nonsensical properties? Could be something like having functors both between Eilenberg-Moore and Kleisli categories, compatible in certain way. Somehow the precise formulation escapes me.

Is anybody aware of some work on these?

I don't know of very much work about these specifically, or a characterization of them in terms of how they act on Eilenberg-Moore or Kleisli categories. But there is a precise sense in which they are the "pseudo" (or "strong") to Street's "lax" morphisms. Namely, there is a 2-monad $M$ on $Cat$ whose algebras are categories equipped with a monad, and these are the pseudo and lax $M$-morphisms respectively.
• I don't know where it is written down, but it has as easy description. A monad on a category $C$ is the same as a monoid in the strict monoidal category $[C,C]$ of endofunctors. The augmented simplex category $\Delta_a$ is the free strict monoidal category on a monoid. Thus, a monad on $C$ is the same as a strict monoidal functor $\Delta_a \to [C,C]$, or equivalently an action on $C$ of the monoid $\Delta_a$ in $\mathrm{Cat}$. Thus, $M(C) = \Delta_a \times C$. – Mike Shulman Jul 28 '17 at 5:52