@Wojowu A comment by Igor Belegradek below the linked MO answer says that the dim $\geq 6$ result is given on p. 107 of Kirby-Siebenmann.

I didn't think you were implying it, just recognizing the situation for what it is. Other people have also been asking me behind the scenes. Thanks for your reaction to Gadella and Gómez!

@WillG I seem to have too many irons in too many fires and lots of stuff doesn't get the follow-through they deserve. However, the paper by G.G. Gould, londmathsoc.onlinelibrary.wiley.com/doi/abs/10.1112/jlms/…, seems to have some careful discussion that rectifies some shortcomings in the classic work of Gel'fand and Vilenkin.

@მამუკაჯიბლაძე To keep things simple and manageable, I'm relying on the fact that every (symmetric) monoidal category is (symmetric) monoidally equivalent to a (symmetric) strict monoidal category, so that I may assume WLOG that $M$ is symmetric strict monoidal. Does this satisfactorily address your question? For some technical details on monoidal strictification, I might point to this section in the nLab, ncatlab.org/nlab/show/clique#monoidal_strictifications. Anyway, to be more accurate, the permutation groupoid $S$ is the free symmetric strict monoidal category on one generator.

@მამუკაჯიბლაძე "When they do coincide there also is the left action that I had in mind but maybe it can also be defined abstractly? Say, in any symmetric monoidal category?" <-- yes. For a symmetric monoidal category $M$, there is a sm category $M^\bullet$ whose objects are functors $M^n \to M$, and whose monoidal product is the obvious composition $M^m \times M^n \to M \times M \to M$. This has a distinguished object $1_M: M \to M$. The permutation groupoid $S$ is the free sm category on one object x. The induced sm functor $S \to M^\bullet$ taking $x$ to $1_M$ gives the desired left actions.

The more sets that you take the intersection over, the smaller the intersection. Or, the fewer sets that you take the intersection over, the larger the intersection.

I must have enjoyed Sergei's comment 5 years ago, because the software shows I upvoted it. But it hits me afresh this morning. Someone ought to collect all the delightful apercu scattered around MO and make a book out of them.

@MartinBrandenburg No, I think I meant exactly one, because you always have a composite $\mathbb{Z} \times F_\alpha \to \mathbb{Z} \to R$.

For the second question, the concept of equicontinuity is relevant. See en.wikipedia.org/wiki/Arzel%C3%A0%E2%80%93Ascoli_theorem

Finally, this survey arxiv.org/pdf/2403.19677 seems to give useful pointers to the theory of linear lattices.

I believe this is really closely related to passages in Freyd-Scedrov which start at 2.156 and continue past 2.157; unfortunately I cannot find my copy of the book. But my memory is that they present a similar sort of graphical calculus for identities in one-object allegories in which composition of morphisms gives their lattice theoretic join, and this is close to the same thing. Someone really ought to look carefully into these connections. There is probably a PhD thesis in there. (All this is also closely connected to Mal'cev theories.)

@MartinBrandenburg This is a hard question to answer, and I am no expert. However, each submodule defines a congruence (an internal equivalence relation in the category of modules) by taking cosets, and these equivalence relations commute under relational composition. Therefore we are in the presence of a linear lattice, in the sense of Rota. The thesis of Marc Haiman, scholar.google.com/scholar?cluster=12728508576024553972, gives a graphical procedure for generating identities that hold in linear lattices. The first of the higher Arguesian identities appears on page 73.

I mean the notation $Sh(i_1, \ldots, i_k)$.

Yes, that's right.

I'm not sure I follow "the induced map on homology". You introduced the functor $C(\mathbb{R}, -)$, but perhaps you really mean to refer to a natural transformation with components of the form $X \to C(\mathbb{R}, X)$, to which you then apply $H_n$ to define this induced map? (I believe the only such transformation is the one taking a point $x \in X$ to the constant function $\mathbb{R} \to X$ valued at $x$.)