Comment by Gro-Tsen on Non-trivial subfield of ${\bf Q}(\sqrt[3]{a+\sqrt{b}})$
@coudy In this case, put an edit in boldface at the top saying something like “this question is badly worded and shouldn't have been posted”, and briefly summarize the problem or the issue making it...
View ArticleWhat topos-theoretic construction lies behind the “symmetric model”...
Suppose we want to prove that (classical!) $\mathsf{ZF}$ does not prove, say, “for every infinite set $A \subseteq \{0,1\}^{\mathbb{N}}$ there exists an injection $\mathbb{N} \to A$” (I take this...
View ArticleComment by Gro-Tsen on Multiplicative cancellation for trivial vector bundles
Since I had “forgotten” why the isomorphism $I⊕J\cong(I⊗J)⊕O_F$ holds, here is a reference: Keith Conrad, “Ideal classes and relative integers”, lemma 8.
View ArticleComment by Gro-Tsen on Beyond the Bring Radical: What is known about...
Hilbert's 13th problem (of which there are conflicting interpretation, so caveat) and the notion of essential dimension seem relevant to mention here, but I don't feel competent enough to say more, and...
View ArticleComment by Gro-Tsen on When can we add choice to a model of ZF
Probably stupid question: what happens in your question if we replace “countable transitive model” by just “transitive model”? Is there a statement $\tau$ such that a transitive model $M$ of ZF...
View ArticleComment by Gro-Tsen on Delta distribution on manifolds
It seems to be that you've convincingly argued that the answer to your own question is no. If there were a “natural” way to define a distribution $δ_p$ for every $p$, then comparing (chartwise) it to...
View ArticleComment by Gro-Tsen on Is there a continuous partition of space into circles?
Wait, does the word “circle” here refer to a submanifold diffeomorphic to $S^1$ or to a bona fide Euclidean circle (set of points on a plane at equal distance from a center point)? Because the question...
View ArticleComment by Gro-Tsen on Internal logic of the topos of simplicial sets
@HDB: Regarding your comment of 2023-04-03 13:13:59Z, it seems to me that an abstract simplicial complex on $[n] := \{0,\ldots,n\}$ is precisely the same thing as a simplicial subset of $\Delta^n$...
View ArticleEvery real function has a dense set on which its restriction is continuous
The title says it all: if $f\colon \mathbb{R} \to \mathbb{R}$ is any real function, there exists a dense subset $D$ of $\mathbb{R}$ such that $f|_D$ is continuous.Or so I'm told, but this leaves me...
View ArticleSimplicial complexes on $[n] := \{0,\ldots,n\}$ that are identical under any...
For $n\in\mathbb{N}$, let us denote by $\Omega(n)$ the set of all (possibly empty) “abstract” simplicial complexes on $[n] := \{0,\ldots,n\}$ (“on $[n]$” means “labeled by the elements of $[n]$”). To...
View ArticleComment by Gro-Tsen on Computing the Second Exterior Power of Certain Ideals...
The fact that your ideals have two generators defines a surjection $R^{\oplus 2}\to I$. Now if you can compute $r$ generators for the kernel of that map (aka “syzygies” between your generators), this...
View ArticleComment by Gro-Tsen on Is the contravariant power set functor more "natural"...
Counterpoint: The covariant powerset functor is a monad in a fairly natural way (the monad's multiplication $\mathcal{P}^2(X)\to\mathcal{P}(X)$ takes a set of subsets of $X$ to their union), and the...
View ArticleAnswer by Gro-Tsen for Examples of common false beliefs in mathematics
I just learned incidentally in the comments of another question here that it is not true that every proper subgroup is contained in a maximal (proper) subgroup. A counterexample is easy to find: the...
View ArticleComment by Gro-Tsen on Implicit uses of Countable or Dependent Choice
It's worth noting in relation to this answer that while the equivalence between sequential and ε,δ-continuity at a point requires some form of Choice, the equivalence between sequential and...
View ArticleGraph chromatic numbers defined by interactive proof
Edit (2020-07-15): Since the discussion below is perhaps a bit long, let me condense my question to the followingShort form of the question: Let $G$ be a finite graph (undirected and without...
View ArticleComment by Gro-Tsen on Possible gaps for a function and its Fourier transform
I don't understand your question: Christian Remling's comment which you referred to points out that if $g$ and $\hat h$ both vanish on a half-line, then $f(x,y) := g(x)\,h(y)$ vanishes on a half-plane...
View ArticleWhat's the deal with De Morgan algebras and Kleene algebras?
The notion of Boolean algebras, and the corresponding classical propositional logic, is very standard, and it is easy to find information about them (for example, among many other such works, there is...
View ArticleComment by Gro-Tsen on Is there an 'unnatural' topological construction of an...
This doesn't answer the question you asked because you seem to care about extensions of the coefficient field $\mathbb{F}_p$, but I think it's at least worth pointing out that, if $k :=...
View ArticleComment by Gro-Tsen on Is there an 'unnatural' topological construction of an...
On the line of “what does the algebraic closure of $\mathbb{F}_p$ look like?”, it is also probably worth pointing out that, for $p=2$, this can be seen as the set of ordinals less than...
View ArticlePreimage of a sublocale by a morphism of locales: description by nucleus?
For completeness of MathOverflow, and to avoid any possible misunderstanding, let me recall the following terminology and facts, which should be standard (experts skip the following 2–3 paragraphs...
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