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arxiv: 2111.14757 · v3 · submitted 2021-11-29 · 🧮 math.AT · math.AG· math.CT

The surface category and tropical curves

Jan Steinebrunner This is my paper

Pith reviewed 2026-05-24 12:22 UTC · model grok-4.3

classification 🧮 math.AT math.AGmath.CT
keywords surface categoryclassifying spacetropical curvesbordismcospan categoriesrational homotopymoduli spacespositive boundary surgery
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The pith

The classifying space of the surface category hBord_2 is rationally equivalent to a circle.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The paper computes the classifying space of the discrete surface category hBord_2, whose objects are closed oriented 1-manifolds and whose morphisms are diffeomorphism classes of oriented surface bordisms. It shows this space is rationally equivalent to a circle, hence much smaller than the classifying space of the topologically enriched Bord_2. For the wide subcategory hBord_2^{χ≤0} that excludes all morphisms containing disks or spheres, the classifying space is instead large, with rational homotopy groups containing the homology of all moduli spaces of tropical curves Δ_g as a summand. The computation rests on proving that positive boundary surgery applies to labelled cospan categories, a class of discrete symmetric monoidal categories, and uses the same method to show that the (2,1)-category of cospans of finite sets has contractible classifying space.

Core claim

We compute the classifying space of the surface category hBord_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 Bord_2 studied by Galatius-Madsen-Tillmann-Weiss. However, we also show that for the wide subcategory hBord_2^{χ≤0} ⊂ hBord_2 that contains all morphisms without disks or spheres, the classifying space B(hBord_2^{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 (2,1

What carries the argument

Labelled cospan categories, to which a version of positive boundary surgery is applied to compute their classifying spaces.

If this is right

  • B(hBord_2) is rationally equivalent to a circle.
  • The rational homotopy groups of B(hBord_2^{χ≤0}) contain the homology of the moduli spaces Δ_g of tropical curves as a summand.
  • The classifying space of the (2,1)-category of cospans of finite sets is contractible.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

  • The appearance of tropical curve homology suggests that discrete bordism categories with restricted Euler characteristic encode geometric moduli problems in their homotopy.
  • Contractibility of the finite-set cospan category may simplify computations for other discrete symmetric monoidal categories built from sets.
  • The contrast between the full surface category and its χ≤0 subcategory indicates that low-genus components dominate the rational homotopy in the unrestricted case.

Load-bearing premise

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

What would settle it

A direct calculation of the rational homotopy groups of B(hBord_2) that finds them different from those of a circle would falsify the main equivalence.

Figures

Figures reproduced from arXiv: 2111.14757 by Jan Steinebrunner.

Figure 1
Figure 1. Figure 1: An example of how the map µ (de€ned in 6.9) can be evaluated on a 4-simplex in B(Cobχ≤0 2 ). ‘e 4-simplex is parametrised by (t0, t1, t2, t3, t4) ∈ [0, 1]5 with Pti = 1. ‘e double-suspension Σ 2∆g is given by triples [(G, w, d), a, b] where a, b ∈ [0, 1] with a + b ≤ 1 and (G, w, d) a stable metric graph of genus g and volume 1 − a − b. ‘is is identi€ed with the base-point if a = 0 or b = 0. To evaluate µ … view at source ↗
Figure 2
Figure 2. Figure 2: A morphism in Cob2 can be thought of as a cospan of €nite sets labelled in N. to section 2, which serves as an introduction to this paper from the perspective of labelled cospan categories. De€nition 1.3. A labelled cospan category is a symmetric monoidal category C together with a sym￾metric monoidal functor π : C → Csp satisfying four axioms, which ensure that every object and morphism in C uniquely deco… view at source ↗
Figure 3
Figure 3. Figure 3: ‘e three conditions that the morphism PA : A → O ⊗ A has to satisfy for all connected objects B, arbitrary objects M, N, and connected morphisms U, V , W. ‡eorem 2.44 (Surgery ‘eorem, see 4.1). If the labelled cospan category (C → Csp) admits surgery, then the inclusion of simplicial sets N•C ∂+ ⊂ Nnc • C induces an equivalence: B(C ∂+ ) ' |N nc • C|. Combining this with the decomposition theorem we have t… view at source ↗
Figure 4
Figure 4. Figure 4: Le‰: an object ((M, W, W0 ),(A, a)) ∈ F0 g=3(Cob2). Middle: the value of the natural trans￾formation α : G ⇒ Id at this object. Right: the value the functor G on this object. Lemma 3.51. For all d ≥ 2 and all di‚eomorphism types [Q] of closed d-manifolds the factorisation category F[Q] (Cobd) has a contractible classifying space. 38 [PITH_FULL_IMAGE:figures/full_fig_p038_4.png] view at source ↗
Figure 5
Figure 5. Figure 5: On top: a 3-simplex W ∈ C nc 3 for the labelled cospan category C = Cob2. Below: ‘e representing space |W| with labels in N recording the genus and a possible surgery path. De€nition 4.9. For W ∈ C nc n a surgery path is a continuous path p : [0, 1] → |W| such that the composite pR : [0, 1] → |W| → [0, n + 1] is piece-wise linear, pR(0) ∈ {0, n + 1}, and pR(1) ∈ {0, 1, . . . , n + 1}. If the integer i := p… view at source ↗
Figure 6
Figure 6. Figure 6: A slide show of the homotopy ρ s α(W, t) for a 2-simplex W as s moves from 1 to 0. Lemma 4.16. Œe construction in de€nition 4.15 yields a continuous map ρ : |C σ • | × [0, 1] −→ |C nc • |, ((W,(p α )α∈A, t), r) 7→ ρ r Ain(W, t) and restricted to r = 1 this map is the standard projection |C σ • | → |C nc • |. Here as before Ain ⊂ A denotes the subset corresponding to those paths that start at 0. Proof. ‘e m… view at source ↗
Figure 7
Figure 7. Figure 7: When applying PA twice the order does not ma‹er. De€nition 4.19. Consider a point (W, t) ∈ |C nc • |, a surgery path p for W, an element α ∈ ΩO disjoint to W, and a parameter r ∈ [0, 1]. ‘en we de€ne the basic surgery of (W, t) along the path p to be Kr (p,α) (W, t) = σ pR(r) pΩ(r),α (W, t). Whenever pΩ(r) is not de€ned because pR(r) ∈ {0, . . . , n + 1} we choose either the most recent well-de€ned value p… view at source ↗
Figure 8
Figure 8. Figure 8: A slide-show depiction of the homotopy Ks p,α(W, t) along the surgery path p indicated in the €rst picture, as s goes from 0 to 1. See remark 4.20. 48 [PITH_FULL_IMAGE:figures/full_fig_p048_8.png] view at source ↗
Figure 9
Figure 9. Figure 9: A slide-show depiction of the homotopy Ks p,α(W, t) along a surgery path p that starts at 1. 49 [PITH_FULL_IMAGE:figures/full_fig_p049_9.png] view at source ↗
Figure 10
Figure 10. Figure 10: ‘e functor S evaluated on a morphism f : (X, U) → (Y, V ). ‘e identi€cations ∂U ∼= {1, 2, 3, 4} and ∂V ∼= {1, 2, 3, 4} are le‰ implicit. 59 [PITH_FULL_IMAGE:figures/full_fig_p059_10.png] view at source ↗
Figure 11
Figure 11. Figure 11: A €ltration induced by a height function [PITH_FULL_IMAGE:figures/full_fig_p063_11.png] view at source ↗
Figure 12
Figure 12. Figure 12: An example of how the map Φ : B(F χ≤0 5 ) → ∆5 from de€nition 6.6 can be evaluated on a 2-simplex. ‘e 2-simplex is parametrised by (t0, t1, t2) ∈ [0, 1]3 with t0 + t1 + t2 = 1. Φ sends this to the weighted metric graph with one vertex per component in each morphism Wi , weighted by the genus of this component, and one edge of length 1 |π0Mi| ti per circle in each object Mi . A‰erwards, all valence 2 and g… view at source ↗
read the original 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.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Referee Report

0 major / 2 minor

Summary. The paper computes the classifying space of the discrete surface category hBord_2 (objects: closed oriented 1-manifolds; morphisms: diffeomorphism classes of oriented surface bordisms) and shows it is rationally equivalent to a circle, hence much smaller than the topological version studied by Galatius-Madsen-Tillmann-Weiss. For the wide subcategory hBord_2^{χ≤0} excluding disks and spheres, the classifying space is larger, with its rational homotopy groups containing the homology of all moduli spaces of tropical curves Δ_g as a summand. The key technical result is a version of positive boundary surgery that applies to labelled cospan categories; this is also used to prove that the (2,1)-category of cospans of finite sets has contractible classifying space.

Significance. If the results hold, the work sharply distinguishes the homotopy types arising from discrete versus topologically enriched bordism categories and exhibits an unexpected summand linking the rational homotopy of a bordism category to the homology of tropical curve moduli spaces. The general positive-boundary-surgery theorem for labelled cospan categories is a reusable tool that may apply to other discrete symmetric monoidal categories; the manuscript ships this new technical result together with its applications.

minor comments (2)
  1. The introduction would benefit from an early, self-contained example of a labelled cospan category before the general definition is stated.
  2. Notation for the subcategory (hBord_2^{χ≤0}) and the tropical summand should be cross-referenced explicitly when the summand statement is first claimed.

Simulated Author's Rebuttal

0 responses · 0 unresolved

We thank the referee for their positive summary of the manuscript, recognition of its significance, and recommendation of minor revision. No specific major comments or criticisms are listed in the report, so we have no points requiring point-by-point rebuttal or revision at this stage. We are happy to address any minor editorial suggestions in a revised version.

Circularity Check

0 steps flagged

No significant circularity; central claims rest on new surgery theorem for labelled cospan categories

full rationale

The derivation chain begins with the definition of hBord_2 and the labelled cospan category framework, then invokes a new technical theorem (positive boundary surgery for such categories) to compute B(hBord_2) ≃_Q S^1 and to extract the Δ_g summand in the χ≤0 subcategory. These steps are presented as consequences of the surgery result rather than reductions to fitted inputs, self-definitions, or prior self-citations. The contractibility of the cospan category of finite sets is likewise derived from the same theorem. External comparison to Galatius-Madsen-Tillmann-Weiss is non-circular. No load-bearing step reduces by construction to its own inputs, and the paper is self-contained against external benchmarks.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

The paper builds on standard axioms from category theory and differential topology, with the main new assumption being the applicability of the surgery technique to the defined class of categories.

axioms (2)
  • standard math Diffeomorphism classes of oriented surfaces form a symmetric monoidal category under disjoint union.
    This is the definition of the surface category hBord_2.
  • ad hoc to paper Positive boundary surgery applies to labelled cospan categories.
    This is stated as the technical key result enabling the main computations.

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Reference graph

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