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

- 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?
- 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:

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.

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.

- I would package this as a tuple,
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.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.