I think he just means that no two of that list of functions are equal (as functions). In other words, that the monoid map $F[x, y] \to \hom(\mathbb{N}, \mathbb{N})$ from the free monoid on two generators to the monoid of endofunctions on $\mathbb{N}$, sending $x$ to $f$ and $y$ to $g$, is an injection.

In any event, the underlying functor $LCH \to Set$ is not monadic (doesn't reflect isos). For example, you can topologize $[0, 1]$ the usual way, or you can topologize it as if it were homeomorphic to $[0, 1) \cup \{2\}$. Both are locally compact Hausdorff. There is an obvious continuous bijection from the latter to the former.

This discussion may be getting a bit arcane, but (David): is that what Paul Taylor was saying in that article? (He does indeed say "LCLoc is monadic" (3.11), but reading the proof and the intro, it looks to me he's really saying that LCLoc^{op} is monadic over LCLoc. Please correct me if I'm wrong.) I would like to hear more from Paul Siegel as well.

Since your interest comes from analysis, can we assume that the spaces of interest are not just locally compact Hausdorff but have other properties as well, for example first-countability?

I didn't see Friedman's sense of "foundationally complete" from the link provided (and actually, I was actively following the FOM discussions during that time). The responses to categorical/structuralist foundations (advocated by McLarty, Awodey, and others) by Friedman, Simpson, and others committed to materialist foundations were, IMO, disappointingly shallow and ridiculously emotional (hostile). The question Friedman raises about V(w+w) is very interesting however (and particularly interesting for the question of adequacy of Mac Lane's set theory!).

I first read Flatland when I was 11 or 12, and to me it was an invitation to imagine encounters with beings who live in more than three dimensions! It was a real eye-opener.

+1 Nice answer, Mike. Now I don't have to struggle with this any more! :-)

Ditto Mike's last sentence. The last example I tried looking at was where $G$ is the 2-dimensional oriental in the category 2-Cat, which I'm pretty sure is not dense (because Street's nerve functor is not full on 2-Cat).

Sorry, Joey -- compact Hausdorff spaces are a cocomplete category. This is well-known. (You are right however that colimits there are not computed as they would be in $Top$.) I don't think however the question has been totally settled by Mike's example, because one can still present a compact Hausdorff space as a colimit of a diagram consisting totally of 1's (essentially because compact Hausdorff spaces are monadic over sets).

The explanation given by Conway and Guy in The Book of Numbers is also in the spirit of these two answers.

I think I did correctly transcribe your notion of right action of an $V$-valued operad $A$ on a $V$-object $X$, namely as a right module structure of the P-rep $X^{\otimes *}$ over the monoid $A$. But never mind: it might be more fruitful to pursue the thread below your own answer. If this is coming from things related to plethysm, I'd be interested.

Well, that's very close to things I've seen before, namely strengths on functors. However, before trying to say more in these damned boxes, I'm having trouble parsing something: a priori, the right side of your arrow lives in sets, and the left-hand side in C. Do you mean the left hand side is the underlying set of the M(Y)-fold coproduct of X? And the argument inside the M on the right is similarly the underlying set of the Y-fold coproduct of X? Anyway, it's good to get more of an idea what's motivating your original question.

I thought "exotic sphere" just meant a differentiable structure on the standard topological sphere which is inequivalent to the standard differentiable structure. So all you have to do is triangulate the standard topological sphere. Am I making a dumb mistake?

But surely the coalgebraic applications of bisimulations and the applications to ZF - Foundation + Anti-foundation are closely connected? In ordinary ZF, sets are constructed recursively, whereas in the alternative set theory using AFA, sets are constructed corecursively. There's a little bit on this at the nLab under the article "axiom of foundation".

Thanks for bringing this to my attention, Francois. I may have something to add a little later.