The surface category and tropical curves

TL;DR

This paper computes the classifying space of a surface category, revealing it is rationally equivalent to a circle, and uncovers a surprising connection between its subcategory and the homology of tropical curve moduli spaces.

Contribution

It introduces the concept of labelled cospan categories and applies positive boundary surgery to analyze classifying spaces of surface categories.

Findings

Classifying space of $h ext{Bord}_2$ is rationally equivalent to a circle.

Subcategory $h ext{Bord}_2^{ ext{chi} extless 0}$ has a classifying space with homology of tropical curve moduli spaces.

A version of positive boundary surgery applies to labelled cospan categories.

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.