The recent paper Calin Tataru, Partial orders are the free conservative cocompletion of total orders. arXiv:2404.12924 has shown that the conservative cocompletion of the simplex category $\Delta$ is the category $\mathsf{Pos}$ of posets. There are a number of other important "categories of geo...

Think of a one-dimensional infinite walk as a map $$w\colon \mathbb{N}\to \{-1,1\}.$$ (If it is more convenient, you can think of a walk as a subset of $\mathbb{N}$, or as a binary word, or as any other convenient model.) Similarly, a finite walk, say of $n$-steps (for some integer $n\geq 1$), ...

A cardinal is Reinhardt if $\kappa$ is the critical point of a nontrivial elementary embedding of $V$ to itself, where $V$ is the class of all sets. As Reinhardt cardinals are inconsistent with $\mathrm{ZFC}$, work in $\mathrm{ZF}j$, which is $\mathrm{ZF}$ with replacement and separation for form...