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The surface category and tropical curves

Jan Steinebrunner

TL;DR

The paper develops a comprehensive framework for analyzing the surface cobordism category via labelled cospan categories, proving that the classifying space $B( ext{Cob}_2)$ is rationally circle-like and that refining to $ ext{Cob}_2^{oldsymbol{ abla}_{oldsymbol{ }}}$ reveals a rich rational homotopy structure connected to tropical moduli spaces $oldsymbol{ riangle}_g$. Central to this is a Decomposition Theorem that expresses $B( ext{Csp}( ext{C}))$ as a fiber sequence over a free infinite loop space built from factorization categories $ ext{F}_g( ext{C})$, and a Surgery Theorem showing that, under suitable admissibility, the positive-boundary data controls the homotopy type. The work unpacks the connection between cobordism categories, cospan categories, and tropical geometry by constructing equivalences to graph- and moduli-theoretic objects, including a rational map to $oldsymbol{ riangle}_g$ and a detailed comparison to $oldsymbol{ riangle}_g$. Collectively, these results provide a unifying, computable route to the homotopy types of cobordism-related classifying spaces and illuminate the role of tropical moduli in the topology of surface categories.

Abstract

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 $\mathrm{Cob}_2^{χ\le0} \subset \mathrm{Cob}_2$ that contains all morphisms without disks or spheres, the classifying space $B\mathrm{Cob}_2^{χ\le0}$ 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 labelled cospan categories. We also use this to show that the $(2,1)$-category of cospans of finite sets has a contractible classifying space.

The surface category and tropical curves

TL;DR

The paper develops a comprehensive framework for analyzing the surface cobordism category via labelled cospan categories, proving that the classifying space is rationally circle-like and that refining to reveals a rich rational homotopy structure connected to tropical moduli spaces . Central to this is a Decomposition Theorem that expresses as a fiber sequence over a free infinite loop space built from factorization categories , and a Surgery Theorem showing that, under suitable admissibility, the positive-boundary data controls the homotopy type. The work unpacks the connection between cobordism categories, cospan categories, and tropical geometry by constructing equivalences to graph- and moduli-theoretic objects, including a rational map to and a detailed comparison to . Collectively, these results provide a unifying, computable route to the homotopy types of cobordism-related classifying spaces and illuminate the role of tropical moduli in the topology of surface categories.

Abstract

We compute the classifying space of the surface category whose objects are closed -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 studied by Galatius-Madsen-Tillmann-Weiss. However, we also show that for the wide subcategory that contains all morphisms without disks or spheres, the classifying space is surprisingly large. Its rational homotopy groups contain the homology of all moduli spaces of tropical curves 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 -category of cospans of finite sets has a contractible classifying space.
Paper Structure (34 sections, 68 theorems, 211 equations, 11 figures)

This paper contains 34 sections, 68 theorems, 211 equations, 11 figures.

Table of Contents

  1. Introduction
  2. Labelled cospan categories
  3. Labelled cospans
  4. Definition
  5. Decomposing labelled cospans
  6. Weighted cospans
  7. The enhanced cospan category
  8. The unlabelled case
  9. The general case
  10. Higher labelled cospan categories
  11. The decomposition theorem
  12. Partial commutative monoidal categories
  13. A general fiber sequence
  14. A base-change theorem for simplicial spaces
  15. The simplicial groupoid D(C)
  16. ...and 19 more sections

Key Result

Theorem 1

The rational homotopy groups of $B(\mathrm{Cob}_2^{{\chi\le0}})$ are where $|\alpha| = 1$, $|\rho_i| = 4i+2$, and $\Delta_g$ is the moduli space of tropical curves of genus $g$ and volume $1$.

Figures (11)

  • Figure 1: An example of how the map $\mu$ (defined in \ref{['defn:mu']}) can be evaluated on a $4$-simplex in $B(\mathrm{Cob}_2^{{\chi\le0}})$. The $4$-simplex is parametrised by $(t_0, t_1, t_2, t_3, t_4) \in [0,1]^5$ with $\sum t_i = 1$. The double-suspension $\Sigma^2\Delta_g$ is given by triples $[(G,w,d), a, b]$ where $a,b \in [0,1]$ with $a+b \le 1$ and $(G,w,d)$ a stable metric graph of genus $g$ and volume $1-a-b$. This is identified with the base-point if $a=0$ or $b=0$. To evaluate $\mu$ we sum over closed components $V$ of the diagram, discarding components with boundary. To each $V$ we assign a stable metric graph with an edge of length $t_i/k_i$ for every time $V$ intersects the $i$th vertical line, where $V$ intersects the $i$th line $k_i$ times. To be precise, valence $2$ genus $0$ vertices should be deleted and the length of their adjacent edges added. The coordinates $(a,b)$ in the suspension $\Sigma^2\Delta_g$ are given by the sum of the $t_i$ "before" and "after" $V$, respectively.
  • Figure 2: A morphism in $\mathrm{Cob}_2$ can be thought of as a cospan of finite sets labelled in $\mathbb{N}$.
  • Figure 3: Left: an object $((M,W,W'),(A,a))\in\mathcal{F}_{g=3}'(\mathrm{Cob}_2)$. Middle: the value of the natural transformation $\alpha:G \Rightarrow \operatorname{Id}$ at this object. Right: the value the functor $G$ on this object.
  • Figure 4: The three conditions that the morphism $P_A:A \to O \otimes A$ has to satisfy for all connected objects $B$, arbitrary objects $M$, $N$, and connected morphisms $U$, $V$, $W$.
  • Figure 5: On top: a $3$-simplex $W \in C_3^{\rm nc}$ for the labelled cospan category $\mathcal{C} = \mathrm{Cob}_2$. Below: The representing space $|W|$ with labels in $\mathbb{N}$ recording the genus and a possible surgery path.
  • ...and 6 more figures

Theorems & Definitions (233)

  • Theorem 1
  • Theorem 2
  • Definition 1.1
  • Definition 1.2
  • Theorem 3
  • Corollary 4
  • Theorem 5
  • Definition 1.3
  • Theorem 6: Decomposition and Surgery Theorem
  • Theorem 7
  • ...and 223 more