The surface category and tropical curves

2021/11/29 by Steinebrunner, Jan
#14T20 #55R40 #57R90 #Algebraic Geometry (math.AG) #Algebraic Topology (math.AT) #Category Theory (math.CT) #FOS: Mathematics

paper · doi:10.48550/arxiv.2111.14757

Abstract

We compute the classifying space of the surface category Cob₂ 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 C₂ studied by Galatius-Madsen-Tillmann-Weiss. However, we also show that for the wide subcategory Cob₂^χ≤0 ⊂ Cob₂ that contains all morphisms without disks or spheres, the classifying space BCob₂^χ≤0 is surprisingly large. Its rational homotopy groups contain the homology of all moduli spaces of tropical curves Δ_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.

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