Non-trivial higher homotopy of first-order theories

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