Comment by Gro-Tsen on Sampling from the uniform distribution on preorderings...
@JackEdwardTisdell I think this doesn't work: the distribution on the partitions defined by the equivalence classes of a random preorder is not uniform: larger classes have a smaller probability of...
View ArticleComment by Gro-Tsen on Symmetry of the Riemannian logarithm
I don't even understand what the statement means: $\log_x y$ is an element of $T_x M$ (the tangent space at $x$) and $\log_y x$ is an element of $T_y M$. Do you mean to imply that you parallel...
View ArticleComment by Gro-Tsen on Why is it so difficult to define constructive...
Answer to a related question on MSE about the cardinality of the set $\Omega := \mathcal{P}(1)$ of truth values. (Beware that I use slightly different definition of “finite”, “subfinite”, etc., than in...
View ArticleComment by Gro-Tsen on Why is it so difficult to define constructive...
@WillSawin Even without CSB, “$X$ injects into $Y$” defines a preordering, and one might imagine redefining “cardinality” as equivalence classes not under “$X$ is in bijection with $Y$” as per Cantor,...
View ArticleComment by Gro-Tsen on Equivalence of definitions of locally finitely...
That $\mathcal{F}$ is “generated by the $s_i$” means precisely that the morphism $\mathcal{A}^{\oplus n} \to \mathcal{F}$ is surjective (see, e.g., EGA 0.5.1.1). This is often expressed, instead, as a...
View ArticleComment by Gro-Tsen on Configurations of injections and surjections (without...
Empty set aside, if there's an injection $f:A\to B$ then there's a surjection $B\to A$ by taking elements of the image of $f$ to their antecedent, and other elements to some fixed element of $A$. (Of...
View ArticleAnswer by Gro-Tsen for Equivalence of definitions of locally finitely...
As pointed out in the comments, the correct definition of “the sections $s_1,\ldots,s_n$ (on $X$, say) generate $\mathcal{F}$ [as an $\mathcal{A}$-module]” is that the morphism of $\mathcal{A}$-modules...
View ArticleOn the history of the “bb” propositional formula that characterizes finite...
The intuitionistic propositional formula $\mathbf{bb}_n$ (in the $n+1$ variables $p_0,\ldots,p_n$) is:$$\bigwedge_{i=0}^n \Big ( \big (p_i \Rightarrow \bigvee_{j\neq i} p_j\big) \,\Rightarrow \,...
View ArticleComment by Gro-Tsen on When do globally generating sections generate the...
That $\mathcal{F}$ is “generated by the $s_i$” means precisely that the morphism $\mathcal{A}^{\oplus n} \to \mathcal{F}$ is surjective (see, e.g., EGA 0.5.1.1). This is often expressed, instead, as a...
View ArticleAnswer by Gro-Tsen for When do globally generating sections generate the...
As pointed out in the comments, the correct definition of “the sections $s_1,\ldots,s_n$ (on $X$, say) generate $\mathcal{F}$ [as an $\mathcal{A}$-module]” is that the morphism of $\mathcal{A}$-modules...
View ArticleComment by Gro-Tsen on Matrices with many traceless powers
The Wikipedia link doesn't give the statement you use. This link (theorem 1 on page 3) has it.
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