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I've split module over a monad off from algebra for an endofunctor. It still needs work, notably the definition of tensor product of bimodules, but it's late and I'm tired.
Also I added a remark to internal category about internal cats as monads in the bicategory of spans. I'm leading up to talking about internal profunctors and 2-sided fibrations, which Mike has been helping me understand at the café.
Thanks. I didn't actually remember that the material had been hiding there.
We might have to equip one entry or other now with links to module over a monad. I just added a link to it into the relevant section at module.
In fact I liked that modules over a monad were considered just a bit richer version of an algebra over an endofunctor; this suggests lots of generalization, of for example distributive laws etc. Regarding that the netries are now separated I added few additional words to have a part of the connection still stated.
Yes, a few words at algebra for an endofunctor would be nice, as would a note at module over a monad.
It looks like you did that, good.
I changed the redirects, but currently algebra over a monad is still redirecting to monad instead of to module over a monad
edit: hm, now it doesn't anymore. false alarm
Added the definition of tensor product of bimodules yesterday.
A new reference (Fiore, Gambino, Kock) at monad, and a complaint in a query box. I think that the idea section needs more general viewpoint than that of algebraic theories, though have difficulty to come up with a canonical one.
While you are still editing monad:
the first sentence of the Idea-section should go something like:
A monad is an endomorphism in a 2-category equipped with the structure of a weak monoid: a 2-morphism from the composite of the morphism to itself, and one from the identity endomorphism to it, satisfying associativity and uniticity.
In the same spirit there is a notion of module over a monad.
Monads are often considered in the 2-category Cat where they are given by endofunctors with a monoid structure on them. Modules over monads in Cat and on Set encode algebraic structures on sets. Therefore modules over a monad are also called algebras overa monad.
What is “weak” about a monad as a monoid?
What is “weak” about a monad as a monoid?
Sorry, that was nonsense.
Under tensor product of modules I added the modifier “reflexive” before the various instances of the word “coequalizer”. For it often happens that composition on one or the other side does not preserve general coequalizers, but it does preserve reflexive coequalizers.
I moved this somewhat obsolete query from monad:
Peter LeFanu Lumsdaine: I did the diagrams with the monad called $(T,\eta,\mu)$ and have only just noticed that that disagrees with what’s used in the preceding description. Was there a particular principled reason for calling it $(A,i,\mu)$ above? I can change the diagrams to agree if so, but if not, might it be easier on newcomers to use $(T,\eta,\mu)$ throughout? Pretty much all the references I know use that as the generic name for a monad. —Peter
Mike Shulman: I like $T$ as the name for a monad. (I also think that as a matter of exposition, this page should start out with monads in $Cat$ and introduce the more general version later, but I don’t have time to implement that right now.)
Peter LeFanu Lumsdaine: I’d been thinking the same; so I’ve re-organised things as you suggest, and added an “idea” section. I think that probably goes into too much detail now, especially since “generalised algebraic theory” is only one of many ideas of what a monad is, but someone else can probably cut it down more dispassionately than I can :-)
Mike Shulman: I don’t think it needs any cutting down. If anything, one could add more description of all the other things that a monad is.
Zoran Škoda: I do not like the idea section. It describes a very special case of monad theory as the principal motivation, namely of monads in the category of sets. I know lots of heavy monad users, including mine, who almost never use monads to describe algebraic theories. For Jon Beck the principal motivation is cohomology theory, for some is the descent theory, for some generalized module theory, for some equivariance, for some relativizing affiness in algebraic geometry…
Ad 8
an endomorphism in a 2-category equipped with the structure of a weak monoid
What weak ? Of course it is a strict monoid in a strict monoidal category of endofunctors.
Edit: i see the answer later in the thread. Sorry.
Added to module over a monad an observation about bimodules as algebras over a composite monad.
The distributive law contains also the condition about the compatibility with the unit. Where did this condition disappear when talking about abstract bimodules ?
An abstract bimodule is in particular a left module and a right module, each of which contains a unit condition. Is that what you’re asking?
No, the distributive law between two monads in Cat has a compatibility triangle between the units of the two monads and the distributive law. If the monads are $S$ and $T$ and $M$ module for both, then the unit $\eta^T_M : M\to T M$ gives a map $S\eta^T_M : S M\to S T M$ and if the distributive law is $d: S T\to T S$ then $d_M \circ S(\eta^T_M) = T(\eta^S_M)$. This is the old case when $M$ is an object. Now I would expect something similar here.
A distributive law doesn’t relate the units of the two monads — there are two triangle axioms, each asserting that the distributive law commutes with two different whiskerings of the same unit, so $d_M \circ S \eta^T_M = \eta^T_{S M}$ etc.
Surely, I was writing the formula in haste without producing the diagram :) Still the question, if the distributive law makes compatibility for a bimodule where an analogy for the unit is lost ?
The particular distributive law constructed on this page does satisfy the unit conditions. You can just check it.
Right, there’s nothing new or tricky going on here. The distributive law is between ordinary monads on an ordinary category, that just happens to be a hom-category of a 2-category, and all of the conditions follow from the naturality of the associativity isomorphism of that 2-category.
I have slightly expanded the warning on terminology in the Idea-section. Now it reads as follows:
Beware that modules over monads in Cat are often called algebras for the monad, since they literally are algebras in the sense of universal algebra, see below. By extension, one might speak of modules over monads in any 2-category as “algebras for the monad”.
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