These are good answers and I have nothing to add on the particular reasons. To sum it up I would say it was impossible to write a comprehensive Elements of Mathematics based on category theory, ...

Answering this question depends strongly on exactly what you want from Higher Topos Theory, because expressing high logical strength is a different goal from expressing an aptly unified logical ...

Pierre Cartier has told me everyone at the time (i.e. everyone in those circles) knew Pierre Samuel wrote the appendix. Incidentally this makes a third person breaking the general rule that all ...

I would be surprised if the purported Grothendieck quote is really his. He does not lean to the short and sweet. It sounds more like an adaptation of another thing Deligne says in "Quelques idées ...

I can tell you how they are related. Before Riemann people would say, for example, the complex square root function (for $z\neq 0$) is two valued, but for any small region of (non-zero) complex ...

Notice that in Poincare's setting there is no base point $x$, since he regards the fundamental group as a group of permutations of fundamental regions -- though he also knew the path interpretation, ...

Proper classes come up when you exhaust the means of forming sets. You need a set when you need to know the means of set theory have not been exhausted -- for example when you want to go on and form ...

As to an example from number theory, Mumford style drawings of arithmetic schemes really helped me to get the difference between split, ramified, and inert ideals in (rings of integers over) algebraic ...

Here I take terminology fromnLab: As long as 2-category means strict 2-category this usage is exactly the same as using $M$ to name both a manifold and its set of points. If $Cat$ is meant to be ...

These bundles have the same sections. But which is simpler depends on your point of view. $X\times \mathbb R\to X$ is simpler set theoretically but is more complicated' topologically in the the ...

Probably $\langle f, g\rangle : C \to A \times B$ is most often just called the arrow to the product. You are right it should not be called a product arrow. People who want a specific name for the ...

On Question (a), yes this is correct. On Question (b), almost yes. If you drop the assumption that ϕ is satisfiable, then Skolem has given a sound and complete procedure for refuting a formula in a ...

In short, algebraic number theory `flies below the radar'' of current Reverse Mathematics. First, after working on this a while, I think it is fair to say most familiar theorems on algebraic ...

I take it you mean sheaves on topological spaces. I think it is valuable to grasp both of two approaches. One (which works essentially the same way for sheaves on sites) is to first see that ...

Infinite models are partly classified by two theorems of $PA^{(-1,2)}$ the subtheory of $PA$ without axioms 1 or 2. Then I will describe the finite models completely. In $PA^{(-1,2)}$ if axiom 1 ...

A discussion of Berkovich spaces and rigid analytic spaces, with references is at http://ncatlab.org/nlab/show/Berkovich+space. For all three, also with references, see http://en.wikipedia.org/wiki/...

In order to be a question about mathematics this would have to ask not when the fact became common knowledge -- since it is already known to bees and dogs as mentioned -- but when it got expressed as ...

I think it will not be easy to find an example with $J=2$. Since a cartesian closed category has finite products, the result cited in David White's answer shows the category $A$ for such an example ...

Ah, Thomas Forster's 1998 paper Forster T. (1994) Weak systems of set theory related to HOL. In: Melham T.F., Camilleri J. (eds) Higher Order Logic Theorem Proving and Its Applications. HUG 1994. ...

So far the answer to the question is no. Lam was speaking in very broad terms, and O'Meara's wish for a direct proof is unmet. When Lam spoke of using the Dirichlet theorem on primes in ...

The distinction is meaningful in many senses. Certainly it is on ZF foundations, and contrary to what Dylan Wilson might seem to suggest, the distinction can be stated formally in the Elementary ...

The comments are all correct. Deligne is citing one half of what is called Belyi's theorem, though experts feel this half was actually clear much earlier due to Weil. Weil used a version of the ...

I must apologize for posting a false answer. in writing up a proof i discovered a gap which grew to a counterexample. In fact not every sheaf of ideals of an etale structure sheaf is finitely ...

I wish there was more direct evidence. For now, it seems likely Bounded Zermelo originated as one unnamed member of a family of theories arising from equiconsistency results. The article Jensen ...

Your example from Witten makes one point: generating functions can make differential operators summarize combinatoric/algebraic information -- basically by making differential operators express ...

The Henselian strategy of Zassenhaus uses $p$-adic bounds rather than the integer exponential bounds of Gelfond's theorem. Weinberger and Rothschild develop this approach over algebraic number ...

It looked straightforward to me, but proof theory can be tricky. Emil's comment convinces me I am not overlooking any pitfall here. PA and EFA interpret the Galois groups of all finite extensions ...