The surface category and tropical curves

Abstract

We compute the classifying space of the surface category $h\mathrm{Bord}_2$ whose objects are closed oriented $1$-manifolds and whose morphisms are diffeomorphism classes of oriented surface bordisms, and show that it is rationally equivalent to a circle. It is hence much smaller than the classifying space of the topologically enriched surface category $\mathrm{Bord}_2$ studied by Galatius-Madsen-Tillmann-Weiss. However, we also show that for the wide subcategory $h\mathrm{Bord}_2^{\chi\le 0} \subset h\mathrm{Bord}_2$ that contains all morphisms without disks or spheres, the classifying space $B(h\mathrm{Bord}_2^{\chi\le0})$ is surprisingly large. Its rational homotopy groups contain the homology of all moduli spaces of tropical curves $\Delta_g$ as a summand. The technical key result shows that a version of positive boundary surgery applies to a large class of discrete symmetric monoidal categories, which we call \emph{labelled cospan categories}. We also use this to show that the $(2,1)$-category of cospans of finite sets has a contractible classifying space.