MathOverflow

$\DeclareMathOperator\odd{odd}$For a natural number $n$ we define $\odd(n)$ to be the remainder, when all factors $2$ of $n$ have been stripped off. Now look only at natural numbers which are in the class of $3$ modulo $4$: $3, 7, 11, \ldots$.
Question: Does there exist a finite cycle with elements $a_1, a_2, \ldots, a_k$, all in the class of $3$ modulo $4$, such that...

Let $\mathcal X$ be a coherent (1-)topos, and let $\mathbf{Mod}_{\mathcal X}$ to be its condensed category of point, defined by
$$ \mathbf{Mod}_{\mathcal X}(S) := \mathbf{Topoi}(sh(S),\mathcal X) $$
for any profinite space $S$. Here I use $sh(S)$ to denote the $\infty$-topos of sheaves on $S$, and I use $sh_{\le 0}(S)$ to denote the 1-topos (but for a 1-topos this makes no difference). Let $\underline{\mathcal S}$ be the small condensed $\infty...

Let $V$ be a locally presentably monoidal category. Is the category of Cauchy-complete small V-categories locally presentable?
Remark: The category of Cauchy-complete small V-categories is by definition the full subcategory of the category of all small V-categories, which is known to be locally presentable. We also know that if $V$ is a locally presentably monoidal category...

Every surface admits metrics of constant curvature, but there is usually a disconnect betweenthese metrics, the shapes of ordinary objects, and typical mathematical drawings of surfaces.
Can anyone give an explicit and intuitively meaningful formulas for negatively curved metrics that are related to an embedding of a surface in space?...