The surface category and tropical curves
@inproceedings{Steinebrunner2021TheSC, title={The surface category and tropical curves}, author={Jan Steinebrunner}, year={2021} }
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…
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Labelled cospan categories and properads
- Mathematics
- 2022
We prove Steinebrunner’s conjecture on the biequivalence between (colored) properads and labelled cospan categories. The main part of the work is to establish a 1-categorical, strict version of the…