Nomenclature help: action vs. module and “pointed” monadic algebras are module objects?

Question

I'm confused about some nomenclature in a paper by J. Goguen from 1975. I'm happy to use definitions however they appear. But I aim to summarise the paper for a non-expert audience and I want them to be able to “Google” modern terminology to find helpful resources. My problem is that I don't know the terminology myself: I'm just a grunt that uses definitions as they are given. I enjoy algebraic taxonomy, but I'm just a tourist.

So any insight you, dear reader, can add would be most appreciated. I'd also like to keep from embarrassing myself to an audience of mathematicians who may read my thesis.

TL;DR

Assume herein that $(𝘾, ⊗, 1_𝘾)$ is a cartesian monoidal category with finite coproducts.

Gougen's main definitions and theorems rest on two isomorphisms: $$ \textbf{Act}^{X^\ast} ≅ \textbf{Mond}^X \quad\text{and}\quad \textbf{Act}_•^{X^\ast} ≅ \textbf{Mod}^X $$ where

• $\textbf{Act}^M$ is the category of (right) monoid actions for an internal monoid $M ∈ \textbf{Mon}_𝘾$
• $\textbf{Mond}^X$ is the category of “$X$-monadic algebras”, which as far as I can tell, are just plain algebras of the endofunctor $(- ⊗ X) \colon 𝘾 \rightarrow 𝘾$. Goguen makes no explicit use of monads, except to invoke their name.
• $\textbf{Act}_•^{M}$ is the category of “pointed” right $M$-actions, which is an $M$-action $S ⊗ M \rightarrow S$ with a global element $1_𝘾 \rightarrow S$.
• $\textbf{Mod}^X$, the category of “pointed” $X$-monadic algebras, which he identifies as $X$-modules.
• $(-)^*$ is a functor from $𝘾$ to free monoid objects in $\textbf{Mon}_𝘾$.

Looking in nLab, a module object is exactly what Goguen calls an $M$-action! (Except that nLab describes a left action.) In fact, nLab says that module objects are alternatively called action objects.

So I'd like to keep the name $M$-action and the category $\textbf{Act}^M_𝘾$, since it is evocative of my intended use. But would like to rename $X$-monadic algebra to $(- ⊗ X)$-algebra (a plain ol' functor algebra) and then a “pointed” $(- ⊗ X)$-algebra is a $1 ⊕ (- ⊗ X)$-algebra.

My concern is that there is some devil in the details. The isomorphisms above are specifically on $X^\ast$, which, of course, can be forgotten back to $𝘾$ to form the basis for a list-monad (once the relevant natural transformations are defined).

Questions

1. Any idea why Goguen uses that definition for a module? In the minds of algebraists, does it invoke some intuition from the algebraic notion of a module that isn't obvious to me?
2. Do my proposed changes in nomenclature make sense? If so, are they an improvement to any audience you could imagine, or am I making too much of things?

The relevant definitions are below.

---

Goguen's definitions

• The specific paper is Goguen's (1975) “Discrete-Time Machines in Closed Monoidal Categories. I”
• Also assume the standard definition of internal monoids/monoid objects giving the category $\textbf{Mon}_𝘾$
• Free monoids $(M^\ast, ⧺, ι₀)$ are defined in the usual way, with $X^\ast = \bigoplus_{n ∈ ℕ} X^{⊗ n}$ along with concatenation ($⧺$) and unit $ι₀$ as the 0-th coproduct injection---shown to witness a free-forgetful adjunction. (I've modernized the notation a bit.)

Now here is the series of definitions that gives me pause:

1. Given $(M, μ, e) ∈ \textbf{Mon}_𝘾$, a right $M$-action is a $𝘾$-arrow $𝜚 \colon S ⊗ M \rightarrow S$ for $S ∈ 𝘾$ and satisfying two commutative diagrams that amount to an association pentagon and unitor triangle that enforce the preservation of the monoid: where $α$ is the associator, $ρ$ is the right unitor of $𝘾$'s monoidal structure. Homomorphisms are $h \colon S \rightarrow S′$ with committing triangles: $α′ ∘ (h ⊗ M) = h ∘ α$, giving the induced category $\textbf{Act}^{M}$.

   • This definition is consonant with modern literature, so I'm satisfied with it.

2. Let $X ∈ 𝘾$. Then an $X$-monadic algebra is a $𝘾$-arrow $δ \colon S ⊗ X \rightarrow S$ with homomorphisms $h \colon S \rightarrow S′$ with commuting rectangles: $δ′ ∘ (h ⊗ X) = h ∘ δ$ giving the induced category $\textbf{Mond}^X$.

   • I would package this as a tuple, drop the monad talk, and call it an algebra of the endofunctor $(- ⊗ X)$. Then, $\textbf{Mond}^X$ is just $(- ⊗ X)\text{-}\textbf{Alg}$, the category of algebras of the functor.
   • I wonder if he calls it monadic because (even if he doesn't say so) the fact that $$\textbf{Act}^{X^\ast} ≅ \textbf{Mond}^X$$ and $(-)^\ast$ induces a monad.

3. A pointed $M$-action is a tuple $(S, α, σ)$ where $α$ is an $M$-action and $σ$ is a global element. Homomorphisms of these additionally satisfy a triangle for preserving the base-point.

4. An $X$-module is a tuple $(X, S, δ, σ)$ where $δ$ is an $X$-monadic algebra and $σ$ is a global element $𝘾(1_𝘾, S)$. Homomorphisms of these additionally satisfy a triangle for preserving the base-points. This gives the category $\textbf{Mod}_𝘾$.

NB:

I am aware of fixpoints of functors, and the relationship that will unfold. That's part of what I'm writing about.

Answer 1

Action versus module

In modern category theory parlance, action and module are typically interchangeable, and their use comes down to author preference: this is a historical artefact of the differing nomenclature of actions for monoids, and modules for rings (categorically, both are instances of the same phenomenon). Generally speaking, my experience is that you are more likely to encounter action in literature motivated by computer science, and module in literature motivated by mathematics. However, there does not appear to be any real reason to prefer one over the other. Therefore, your usage of $M$-action is completely standard.

In algebra generally, precedence has been given historically to more well structured objects (e.g. groups over monoids), leading to terminology like semigroup and nonunital algebra. Gougen's terminology choice is inspired by a blend of algebraic nomenclature, using action to refer to less structured objects, and module to refer to more structured objects. He mentions this choice in the introduction, referring to the modules in the work of Kalman, and contrasting them with semimodules (i.e. nonpointed actions).

Personally, I find using both action and module with different meanings potentially confusing given the typical usage in category theory. I would suggest the terminology pointed $M$-action instead (which Gougen uses himself, informally). While it's true that these are the same as algebras for the endofunctor $I + ({-}) \otimes X$, I would suggest that pointed action is more readable than the mathematical notation.

(As a side note, note that it is generally preferable to use $+$ for the coproduct, rather than $\oplus$, which is often taken to denote an arbitrary monoidal structure, possibly distributing over $\otimes$.)

Monadic algebras

For a monoid $M$ in a monoidal category $\mathscr C$, $({-}) \otimes M$ carries the structure of a monad, sometimes called the action monad. It is the monad associated to the free–forgetful adjunction between the monoidal category and the category of (right) $M$-actions. In other words, the $M$-actions are the algebras for the monad $({-}) \otimes M$.

For an object $X$ in $\mathscr C$, we can also form the category of (right) $X$-actions, which are precisely the algebras for the endofunctor $({-}) \otimes X$. When $\mathscr C$ admits the construction of free monoids $({-})^*$ (i.e. when the forgetful functor $\mathrm{Mon}(\mathscr C) \to \mathscr C$ admits a left adjoint), the forgetful functor from the category of $X$-actions to $\mathscr C$ also admits a left adjoint, and is furthermore monadic. The induced monad is given by $({-}) \otimes X^*$. In other words, the $X$-actions are the algebras for the monad $({-}) \otimes X^*$. This kind of monad is known as an algebraically-free monad, because its algebras are algebras for an endofunctor.

Presumably this monadicity result is why Gougen uses the terminology $X$-monadic algebra. However, the terminology is not standard, and I personally do not find it evocative. Your proposal of $({-} \otimes X)$-algebra is completely sensible. Alternatively, you could simply use $X$-action: in this case one relies on context to disambiguate between the actions for an object and the actions for a monoid. However, this is the same as when one variably talks about the algebras for an endofunctor and the algebras for a monad.