Docfix: Bound instances are right monad modules

[arkeet]

(>>>= id) gives a natural transformation t m . m ⇒ t m, i.e. m acts on the right.

[phadej]

I see, so https://www.sciencedirect.com/science/article/pii/S0890540109002405 has a mistake, what they call left is actually right. (compare to e.g. https://en.wikipedia.org/wiki/Module_(mathematics))

• André Hirschowitz, Marco Maggesi: Definition 8 (Left modules). A left R-module is given by a functor M : C → D equipped with a natural transformation μ: M · R → M, called the action, which is compatible with the monad composition, more precisely, we require that the following diagrams commute...
• Algebra: A left R-module M consists of an abelian group (M, +) and an operation ⋅ : R × M → M such that for all r, s in R and x, y in M, we have: ...

So this fix is correct, @RyanGlScott, @ekmett, do you agree?

[phadej]

Wait. Isn't it that m is the right module of t m. We have t m (m a) -> t m a.

As I said, it's confusing.

[arkeet]

The monad is m and the module is t m. Let me write f = t m. Referring to the definition in nLab, the 1-cell is the functor f: Hask -> Hask and the 2-cell is the natural transformation f . m ⇒ f (i.e. a coherent family of functions f (m a) -> f a). In other words, m acts on f on the right, making t m a right m-module. (The fact that f = t m is parameterized by m is irrelevant and possibly added to the confusion.)

By contrast, a left m-module is a functor f with a natural transformation m . f ⇒ f with some laws, giving a coherent family of m-algebras m (f a) -> f a.

[phadej]

R = m, M = t m. Who is wrong, Maggesi or nLab ?

[benjamin-hodgson]

I think nLab (and @arkeet's fix) is right. Corroborated in https://research-information.bris.ac.uk/ws/portalfiles/portal/70069065/20.pdf and https://hal.archives-ouvertes.fr/hal-02338144v3/document (the latter being from Maggesi's coauthor on the paper you linked above).