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Conway had an analysis of the notorious Steiner-Lehmus theorem, arguing that no "equality-chasing proof" is possible. MO user Timothy Chow initiated a discussion about Conway's analysis on the FOM lis …

This may be somewhat obliquely along the lines you are asking about, but I think it's interesting enough that it deserves to be made public. My friend James Dolan has been developing with a number o …

First, an answer to Pete Clark's comment on the Chinese remainder theorem can be found in Floris Ernst's 2004 University of Otago Master's thesis Multiplicative ideal theory (pdf link). Prüfer domains …

Note: Fedor and Vladimir have already answered the question, but this is a partial answer in the other direction, under a stronger hypothesis. (This answer, which I had earlier deleted, has been edite …

I would motivate them as follows: if topological spaces were invented to give a general meaning to "continuous function", then uniform spaces were invented to give a general meaning to "uniformly cont …

Another categorical example: the laws of arithmetic, as applied to the arithmetic of finite integers, are ultimately explicable by the fact that the category of finite sets $\mathbf{Fin}$ is a cartesi …

It's called the kernel or kernel pair of $f$. It is used all over the place in category theory, for example to describe the useful notion of regular category where one sets up Galois connections which …

Some magic words for this question are "moduli space" or "moduli stack". In the early days, one was interested in a variety or variety-like object which would classify projective complex curves (compa …

Connes was a member, according to M. Mashaal "Bourbaki: A secret society of mathematicians", AMS 2006 (translated from the French by A. Pierrehumbert). It says so on page 18; see the link. But, as wa …

Probably you had some trouble finding this because the search term $2$-$\text{Cat}$ is not accurate enough; you want not the cartesian monoidal product on $2$-$\text{Cat}$, but rather what is called t …

Of course, there Tom Leinster's book Higher Operads, Higher Categories, and there's also a lot of stuff on the nLab. Also see Max Kelly's seminal paper (I believe unpublished until recently), On the o …

I believe that's called the Milgram bar construction: R.J. Milgram, The bar construction and abelian $H$-spaces, Illinois J. Math. 11 (1967), 242-250.

According to unpublished notes by Gavin Wraith ("Notes on arithmetic universes and Gödel incompleteness theorems" (1985)), PRA can be described as an equational theory or as a Lawvere theory, and is a …

What you are describing is an example of Max Kelly's notion of club, closely connected with the concept of operad. The original references date back to the 70's; one reference is G.M.Kelly. On club …

In 2-category theory, there is a notion of "lax idempotent 2-monad" $M$ on a 2-category, for which the multiplication $m: M M \to M$ is left adjoint to the unit $u M: M \to M M$. A typical sort of exa …