This may seem like self-promotion, but: ncatlab.org/nlab/show/Trimble+n-category

@YemonChoi I meant Turbo.

So we must be given already a permutation representation of $B$ that takes $\pi$ to a permutation $i \mapsto \pi(i)$? It's a little confusing, but it looks to me that you are making some identification between $\{1, 2,\ldots, n\}$ and the elements of $B$. Also: shouldn't the particular case described below the fold be $(A, B) = (\mathbb{Z}_2, \mathbb{Z}_4)$?

Okay, Turbo, I have read through the exchanges including the deleted comments. Gerhard and others were quite right. Moving forward: if you post a question and an answer reveals that there was a problem with the question, then let me suggest that the proper course of action is to admit openly that the answer was quite right, the question had a mistake, thank the user for pointing it out, use the strikethrough command so as not to cover up what happened, and then amend the question. People who are trying to be helpful deserve a gracious response and acknowledgment.

I don't know what "moderator taboo" signifies, but FWIW, one thing we've noticed is the huge number of self-deleted posts coming from this and allied accounts, which (cumulatively speaking) is tantamount to self-vandalism, and is frustrating to users who are trying to interact.

@GeorgesElencwajg Please excuse my writing again (I deleted my earlier comment), but there is not now and never has been a problem with pen names. But you should be aware that (the now self-deleted) account was operated by a problem user who runs many, many other accounts and who is playing various games with the site. The "politeness" you see is a veneer; I have begun to document problems I'm seeing in this thread: meta.mathoverflow.net/a/4251/2926 (so far only a very partial list). Please rest assured that I do not say any of this lightly.

Predicativists like Nik Weaver would agree with "madness".

@Qfwfq As a non differential topologist but as a category theorist, I found it obvious.

@SándorKovács Maybe you didn't follow that link and read through the original post and the answers to it, but I am convinced that Asura Path (and other users in his bounty ring -- see his profile, and its peculiar wording which reflects on my edit to the OP there) were created a few days ago in reaction to some of that discussion, and the account is controlled by the same person or small group behind many or most (all?) of these other accounts. I believe these are things the community should beware of before deciding to interact with this user, although you are free of course to ignore them.

Among the interesting things in the Arnold Miller paper cited by Ashutosh is: "Tall remarks that Booth (1968, unpublished) proved that MA [Martin's Axiom -- TT] implies the closed unit interval is not the union of less than $|2^\omega|$ disjoint nonempty closed sets..."

A great book which deserves to be well-known.

Question 3: obviously morally correct, and obviously completely vague! :-) "Pathology" of course is to some degree in the eye of the beholder. The example of o-minimal structures (which I like, by the way) could be a kind of "monster-barring" pursuit, to use phraseology of Lakatos, at least when considered in the light of hopes and dreams of section 5 of Grothendieck's Sketch of a Program.

(I've edited the previous comment in response to a flag.) Which part needs explaining? Do you know what a symmetric monoidal ($k$-linear) functor is? Do you understand that if a concept (such as the notion of Hopf algebra object) is definable in the language of symmetric monoidal $k$-linear categories, then a symmetric monoidal $k$-linear functor will map models of that concept in the domain category to ones in the codomain?

The title was changed to something I disagree with, because "exponent" refers to a quantity denoted by a superscript, and we are not concerned with inequalities between these, but rather with values of exponentiation as a binary operation on cardinals. Thus, I think the original title was more correct than the so-called "improvement" by user64494 -- even if the plural of "exponentiation" sounds awkward to some ears.

Thanks for confirming your identity! Since my time is limited, I choose on a case by case basis whether to respond to comments of people who I think are not acting in good faith. Maybe you can ask Community Management to look into your questions?

@YCor In a case like this, where there is no mistake and obviously no aggression, the preferred spelling of the OP should be respected.

meta.mathoverflow.net/questions/4200/flood-of-similar-new-us‌​ers