MathOverflow

This is a follow-up to my previous question.
In the answer to the above, the following was demonstrated. If $P:\mathbb Z^n\to\mathbb Z^N$, $1<n\leq N$, is a polynomial map such that $DP(x)$ has full rank at every $x\in\mathbb Z^n$, then each fiber $P^{-1}(b)$ of $P$ is finite....

Let
$T(n,m,k)$ be the family of integer coefficients such that $T(n,m,k) = \det(M_n)$ where $M_n$ is the $n \times n$ matrix with $(i,j)$-th element ${i+m \brace j+k}$.
$R(n,m,k)$ be the family of integer coefficients defined for $1 \leqslant k \leqslant \binom{n+m-1}{m}$ such that $R(n,m,k)$ is the product of the elements of the $k$-th composition of $\{ 1,2,\dotsc,n+m-1 \}$ of the size $m$ with elements $e_i$ such...

Does the category $\mathbf{FreeAb}$ of free abelian groups have $\aleph_1$-filtered colimits?
See the nLab for the definition of a $\kappa$-filtered colimit. Here are some thoughts and related observations on this question.
By Does the category of free abelian groups have sequential colimits? it does not have filtered colimits. But having $\aleph_1$-filtered colimits is a strictly weaker property....

It is well-known that there exist infinitely many equilateral triangles whose vertices lie on the edges of a given triangle, and infinitely many squares whose vertices lie on the edges of a given quadrilateral. Motivated by these observations, I would...