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I’ve come across a category $\mathcal{C}$ recently with an object $T$ such that any other object $X$ has a map $f:X\rightarrow T$, and for any two maps $f,g:X\rightarrow T$, there exists a (not necess …

Let $M$ be a compact Riemannian manifold with geodesic distance function $d$, of (normalised) diameter $1$. Some of my favourite manifolds $M$ have the property that there exists an integer $k$ such t …

$\newcommand{\Grp}{\mathrm{Grp}}$Consider the category of groups $\Grp$, and within it we have the solvable groups $S$. Then any group $G$ determines the functor from solvable groups: $$h_G:=\text{hom …

Consider a family $\mathfrak{F}$ of $k$ element subsets of $\{1,2,..,n\}$, where $n\geq 2k$, such that any two members of $\mathfrak{F}$ have nonempty intersection. The Erdos-Ko-Rado theorem asserts t …

Yes, this identity holds, and both sides are equal to $-D^p(x^p)$. We need the following facts, all of which are straightforward to verify, where $D$ is an arbitrary derivation, still working over ch …