I don't know of any general such criterion. A trouble is that considering the fibers abstractly on their own loses the information of how they are "woven together" via the underlying-set functor. But an enhanced criterion in this vein holds for fibrations, where you could say that two fibrations over $C$ are concretely equivalent if their associated fiber functors $C \to Cat$ are naturally equivalent.

I didn't downvote either, but it's very often the case that asking a whole bunch of questions results in the uncomfortable situation where no one answer answers them all, so that no answer should be the accepted answer. This is perhaps not fatal, as long as someone is willing to collate the individual answers later into a summary comprehensive answer (perhaps made CW to preserve decorum) which can then be accepted. But putting lots of questions into a post is probably better avoided in general.

Yeah, except when it isn't. You need the element to be invertible in the algebra.

@Kostya_I Yes, thanks very much, but let's perhaps try to be generous as Matt F. suggests.

Any idea for $1/e$?

To people voting to close: is the answer so obvious? Let's discard the smart-aleck solution $a_0 = 1, a_i = 0$ for $i \geq 1$.

"Endo-operator" sounds very acceptable to me. (Not important here, but in the jargon I am familiar with, we say an arrow is epic, or that it is an epi, and not that an arrow "is epi". I cannot swear how universal this usage is, but if I found myself saying "is epi", I would regard it as a minor slip.)

I'd say go ahead and post details (it would save me from having to write it up myself).

Thanks for clarifying!

I cannot make out what you are alleging in your final sentence, "I also think that Gauss was everything else but dishonest."

Gauss, true to form, did not behave all that graciously with regard to Bolyai, saying in a letter to Farkas Bolyai (formerly a schoolmate of Gauss), in effect, "I cannot bring myself to praise this accomplishment of your son Janos, because to do so would be praising myself". He ends that letter on a slightly better note, but there is no doubt that the letter was received as a wet blanket on the young man's enthusiasm. But "stealing" is wrong: Gauss had already worked out detailed calculations on hyperbolic geometry and had satisfied himself that it was wholly consistent.

That said, there is no doubt that Gauss was very much occupied with this question since the time he was a young teenager, and he spoke in some detail about his ideas regarding this question in letters to others. So I think it's a case of more than "only claiming".

I have undeleted this question because the contest ended several days ago. See discussion at meta here: meta.mathoverflow.net/questions/4292/…

Sorry, but what does "admits a topos of presheaves" mean?

I guess @aginensky means this article? mathcs.emory.edu/~brussel/Scans/mumfordpicard.pdf

@AlecRhea If you were thinking of posting an answer about this, I think you should go ahead. It surely might be of interest.

@AlecRhea I hope Chris Heunen wouldn't mind my saying that this precise question is an active point of interest, according to a chat we had at the recent CT meeting.