Hm, you may need to look up what "imperative" means. When you tell someone to add an answer, or to delete a comment, or when you demand a moderator reply, without ever using the magic word "please": those are imperatives. Again, I would strongly urge you to avoid writing in that mode. I'll also say that people leave comments sometimes for readers in general, and not only for you -- you can think of your question as not really owned by you, but now part of the public commons. Thus, even if you don't consider Timothy Chow's comment useful to you personally, others might find it useful.

Advice: avoid the imperative mode when trying to get people to help you, and make a good attempt to answer your own question beforehand, rather than demand that distinguished mathematicians do your work for you (as Carlo Beenakker has done).

You seem to be in the habit of making demands on people. I am not going to comply with your demand that I delete Timothy's comment, because it was made in good faith and intended to be helpful (it is not a call to close as duplicate), and indeed is potentially useful. (Moreover, you seem not to have understood why the linked question was closed as "no longer relevant".) I am complying with your demand to reply because your flag doesn't offer the option of replying privately. As noted by Gerry, you have exhibited little research effort, making the question not a very good one.

Yes, if you instead take the objects to be group elements, and the underlying category to be discrete. David's comment is closely related to the Eckmann-Hilton argument: ncatlab.org/nlab/show/Eckmann-Hilton+argument

@LorenoHeer I understand; I'm stating a general guideline, but you are free to continue as you see best.

@LorenoHeer It's generally not a good idea to keep changing the question as responses come in. This usually winds up being frustrating to the community.

@IvanDiLiberti Sorry for the delay. It's Anders Kock's paper on when multiplication structures are left adjoint to units (aka KZ monads): Journal of Pure and Applied Algebra 104 (1995), 41-59.

How could the equality $K_{\max} = \frac{(N+1)^2 + 1}{2}$ be achieved when $N$ is odd, since the right side is non-integral?

I would put a slightly different spin on Alexandre's advice (which I sort of agree with): it's often in the midst of battling a problem tooth and nail that it becomes much clearer what one needs to know to make further progress. It's that "need to know" that can drive one to the right books to consult. Without this, it can be hard knowing where to invest one's energies as to what to read, with the thousands and thousands of books out there.

I'll have to think harder before I feel confident giving an answer, but I suspect many examples are given by weakly cartesian monads, for example analytic monads coming from operads, and the ultrafilter monad.

For a group $G$, here $BG$ means the category with one object and with the hom given by $G$.

Which toposes are not ordinary spaces? Almost all of them. More or less, the ones which "are spaces" are called localic toposes; they do play an important role however (see for example An Extension of the Galois Theory of Grothendieck, which goes into the details of Dmitri's answer).

By sheer coincidence, I've been listening to Bloom Ascension by Steve Roach while reading this thread, and the album cover looks very similar to the Sylvia C image: steveroach.bandcamp.com/album/bloom-ascension.

@HaoS Sure, those are good points (some punctuation would make them easier to read). But I think you misinterpreted the point I was making, since the importance of logical rigor is something I absolutely believe in. "Intuition" (which may be just another word for "distilled experience") is not enough to catch all mistakes, far from it.

I don't know that Lamport has done a careful study to justify the 33 percent. I sort of doubt it. This is not to deny that significant mistakes do slip by from time to time, and I think it's not a bad idea to secure prized results, such as those in say the theory of finite simple groups, by super-careful methodologies such as fully formalized proofs checked by computer-based proof assistants. This was done for example in the case of the Feit-Thompson theorem.