Let $(S,\eta,\mu)$ be a monad on a category $C$, and $(T,\eta,\mu)$ a monad on a category $D$. The following kind of gadget is ubiquitous: a functor $F:D\to C$, together with a natural map $\sigma: SF\to FT$, satisfying the identities

  • $F\eta = \sigma\circ \eta F$, and
  • $\sigma\circ \mu F = F\mu\circ \sigma T\circ S\sigma$.

I might call this a "functor between monads". Question: What do people actually call these, and what are the standard references?

Some random examples:

  • $S$ is the free commutative ring monad on abelian groups, $T$ is the free commutative monoid monad on sets, and $F$ is the free abelian group on a set. (In this case, $\sigma$ is an isomorphism.)

  • $S=T=J$, the free associative monoid on pointed spaces (where we force the basepoint to be the unit element), with $F=$ loop space. (Enriched category theory provides a lot of examples like this.)

  • If $F=$ identity functor, we recover the usual notion of morphism between monads.

Some notes:

  • A functor between monads as above gives you a way to turn a $T$-algebra into an $S$-algebra, so you obtain a functor $F^*: \mathrm{Alg}_T\to \mathrm{Alg}_S$.

  • You can compose "functors between monads", so there is a category with these as morphisms, and monads as the objects.

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    $\begingroup$ +1 for the examples! $\endgroup$ – Tim Campion Jan 14 '14 at 20:33
  • $\begingroup$ If the forgetful functors for the algebras $S$ and $T$ are fully faithful and $F^*$ is an equivalence of categories, can we deduce that $F$ is also an equivalence of categories? $\endgroup$ – user12580 Jul 4 '18 at 10:32

As far as I know, the first paper on this was:

Ross Street, The formal theory of monads. Journal of Pure and Applied Algebra 2 (1972), 149-168.

He called them monad functors. For the same thing but with the direction of the natural transformation reversed, he called them monad opfunctors. You can also consider the case where the natural transformation is an isomorphism, or even the identity, but I forget what he called them.

(If Street's wasn't the first paper on this, it was certainly early and very influential. And it's a beautiful paper.)

In a bid to try to make the terminology of 2-category theory more systematic, I called them lax maps of monads in my book Higher Operads, Higher Categories (CUP, 2004), and similarly colax etc. I don't know whether anyone else followed suit.

Recently I examined the thesis of one of Street's students, who I think used different terminology from what's in The formal theory of monads. I forget what it was, but I'll look it up if I remember.

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    $\begingroup$ viewing monads as lax functors $1 \to \mathbf{Cat}$, then the 1-cells of the lax functor category $Lax(1,\mathbf{Cat})$ are lax maps of monads (guess this was the reason for Tom's choice of terminology); $Lax(1,\mathbf{Cat})$ yields $\mathbf{Mnd}_\mathbf{Cat}$ as defined by Street. nLab ref for this $\endgroup$ – Eduardo Pareja Tobes Mar 24 '12 at 19:05
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    $\begingroup$ I've used your terminology, Tom. Another good reason for calling these "lax morphisms of monads" is that there's a 2-monad on Cat whose algebras are "categories equipped with a monad", and whose lax morphisms are these. $\endgroup$ – Mike Shulman Mar 26 '12 at 5:25
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    $\begingroup$ It's a little confusing, though, because what Charles described is a lax morphism of monads from T to S, despite the fact that if F is the identity functor, it reduces to what looks like a morphism of some sort from S to T. $\endgroup$ – Mike Shulman Mar 26 '12 at 5:26

Pumplün, D. Eine Bemerkung über Monaden und adjungierte Funktoren. Math. Ann. 185 (1970). 329-337. also defines a category of monads.


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