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@RobertIsrael Funny, but it is already clear from 1 and 3 that the OP is considering present tense as an option, as in "In [2], Roe establishes the following statement" -- certainly not an uncommon construction. There is present perfect, as in "Roe has shown in [2] that the following holds".

This is fantastic!!!!! (Read the link if you didn't already know why.) Honestly, this looks like a winning answer.

Well, one of the very first things that comes to mind that's sort of in this vein is that $\beta X = \hom_{\text{Bool}}(\hom_{\text{Set}}(X, 2), 2)$. But if you want to pursue your analogy at a deeper level, try golem.ph.utexas.edu/category/2012/09/…, where both the ultrafilter monad and the double dualization monad are reckoned to be codensity monads induced by the full inclusions of finitary objects.

Meta discussion: meta.mathoverflow.net/questions/4125/… (I request voting up this comment for visibility). I took the liberty of editing the question to implement some suggestions there.

@NoahSchweber From my end, I have to manually delete comments under the post -- actually I could have also deleted their appearance under the post automatically, but wasn't sure what was desired -- for example, maybe keep the first few but remove the rest. Thoughts?

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@user64494 It is arguably not appropriate to answer a question that you feel does not belong, because by so doing you are reinforcing the idea that inappropriate questions will be answered anyway.

Fosco, $\Delta$ (the augmented simplex category) is initial among strict monoidal categories equipped with a monoid. So given a monad $T = UF$, i.e., a monoid in the endofunctor category $[M, M]$, we get an induced cosimplicial object $\Delta \to [M, M]$, and dually we get a simplicial object $\Delta^{op} \to [N, N]$ which upon evaluation at $x \in Ob(N)$ gives a simplicial object $\Delta^{op} \to N$.

Joseph, I'm not sure what the best solution is. Your proposal might work, but my gut feeling (and I say this as a non-set-theorist) is that probably any one of these questions, say 1. to begin with, is going to be hard and would probably make a good MO question on its own. I'd actually be pleasantly surprised if someone at MO could answer such a question, as I would bet that among active MO members, you are currently the number 1 expert on these sorts of issues.

@BKSpurgeon The same reason any of these are amusing: the absurdity of using such elaborate reasoning to deduce such simple facts.

There can be a problem with asking so many questions in a single post; a general rule of thumb is that this should be avoided, and that asking questions one at a time is usually better. It's true that this rule of thumb is often flouted, but one problem is that no one answer is likely to address all the questions (cf. the comments under this post: mathoverflow.net/a/15857/2926), and another is that such questions lend themselves to long discussions which SE software is meant to discourage. (No doubt there are differing views on the seriousness of these problems.)

Minkov, I agree with Nawaf: he took the time to answer your question, and to change the question so as to render his answer now obsolete (or out-of-date as you put it) is not good form. I recommend that you accept his answer, and ask your part (iii) as a new question, with a bounty if you like.

Maybe that means "explicit" is not an optimal word choice, either as a word or a concept.

Essentially anything written by Milnor is worth reading.

Please: if you have a solution to your question, then that is what should be posted here. Not an assurance that you know how to do it; that is of no use to the community. Finally, the last sentence smacks of sarcasm.