Non-trivial higher homotopy of first-order theories

Abstract

Let $T$ be the theory of dense cyclically ordered sets with at least two elements. We determine the classifying space of $\mathsf{Mod}(T)$ to be homotopically equivalent to ${\mathbb{CP}}^{\infty }$. In particular, ${\pi }_2(|\mathsf{Mod}(T)|)=\mathbb{Z}$, which answers a question in our previous work. The computation is based on Connes' cycle category $\Lambda$.

arXiv comment6 pages