The surface category and tropical curves
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arXiv
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| Format: | Preprint |
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2021
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| _version_ | 1866918439629619200 |
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| author | Steinebrunner, Jan |
| author_facet | Steinebrunner, Jan |
| contents | 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 $h\mathrm{Bord}_2^{χ\le 0} \subset h\mathrm{Bord}_2$ that contains all morphisms without disks or spheres, the classifying space $B(h\mathrm{Bord}_2^{χ\le0})$ 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 \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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2111_14757 |
| institution | arXiv |
| publishDate | 2021 |
| record_format | arxiv |
| spellingShingle | The surface category and tropical curves Steinebrunner, Jan Algebraic Topology Algebraic Geometry Category Theory 57R90, 55R40, 14T20 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 $h\mathrm{Bord}_2^{χ\le 0} \subset h\mathrm{Bord}_2$ that contains all morphisms without disks or spheres, the classifying space $B(h\mathrm{Bord}_2^{χ\le0})$ 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 \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. |
| title | The surface category and tropical curves |
| topic | Algebraic Topology Algebraic Geometry Category Theory 57R90, 55R40, 14T20 |
| url | https://arxiv.org/abs/2111.14757 |