We compute the classifying space of the surface category \(h\textrm{Bord}_2\) 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 \(\textrm{Bord}_2\) studied by Galatius–Madsen–Tillmann–Weiss. However, we also show that for the wide subcategory \(h\textrm{Bord}_2^{{\chi \le 0}}\subset h\textrm{Bord}_2\) that contains all morphisms without disks or spheres, the classifying space \(Bh\textrm{Bord}_2^{{\chi \le 0}}\) 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 labelled cospan categories. We also use this to show that the (2, 1)-category of cospans of finite sets has a contractible classifying space.
