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, $\pi _2(\lvert \mathsf {Mod}(T)\rvert )=\mathbb {Z}$, which answers a question in our previous work. The computation is based on Connes’ cycle category $\Lambda $.

We show that the Lascar group $\operatorname {Gal}_L(T)$ of a first-order theory T is naturally isomorphic to the fundamental group $\pi _1(|\mathrm {Mod}(T)|)$ of the classifying space of the category of models of T and elementary embeddings. We use this identification to compute the Lascar groups of several example theories via homotopy-theoretic methods, and in fact completely characterize the homotopy type of $|\mathrm {Mod}(T)|$ for these theories T. It turns out that in each of these cases, $|\operatorname {Mod}(T)|$ is aspherical, i.e., its higher homotopy groups vanish. This raises the question of which homotopy types are of the form $|\mathrm {Mod}(T)|$ in general. As a preliminary step towards answering this question, we show that every homotopy type is of the form $|\mathcal {C}|$ where $\mathcal {C}$ is an Abstract Elementary Class with amalgamation for $\kappa $-small objects, where $\kappa $ may be taken arbitrarily large. This result is improved in another paper.

People with psychosis face a life-restricting and life-shortening epidemicof obesity, diabetes and cardiovascular disease. This can be predicted bythe associated antecedent risk factors evident from early in psychosis, yetremain largely ignored. Greater coordination between primary care, secondarycare and public health to systematically prevent and intervene earlier forthese physical illnesses offers a realistic solution to reduce this healthinequality.