  We compute the classifying space of the surface category $\mathrm{Cob}_2$
whose objects are closed $1$-manifolds and whose morphisms are diffeomorphism
classes of 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 $\mathcal{C}_2$ studied by
Galatius-Madsen-Tillmann-Weiss. However, we also show that for the wide
subcategory $\mathrm{Cob}_2^{\chi\le0} \subset \mathrm{Cob}_2$ that contains
all morphisms without disks or spheres, the classifying space
$B\mathrm{Cob}_2^{\chi\le0}$ 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 labelled cospan categories. We also use this to show
that the $(2,1)$-category of cospans of finite sets has a contractible
classifying space.
