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Excision of Skein Categories and Factorisation Homology

Juliet Cooke

TL;DR

The paper proves that Walker–Johnson-Freyd skein categories satisfy excision, enabling a description of skein categories as k-linear factorisation homology: Sk_V(Σ) ≃ ∫_Σ^{Cat_k^×} 𝒱. It then relates Tambara’s relative tensor product for k-linear categories to the bar-construction approach, showing the two notions of relative tensor product coincide in Cat_k^× and establishing excision for skein categories in the factorisation-homology sense. Building on this, the authors connect skein categories to the LFP_k framework, demonstrate that free cocompletions of skein categories realize LFP_k factorisation homologies, and apply these results to quantised character varieties, proving that skein algebras for generic q yield deformation quantisations of character varieties. The work unifies skein theory with factorisation-homology, providing a robust method to quantise character varieties via skein-theoretic and categorical constructions. The results generalise known cases (e.g., SL_n character varieties) and offer a broad framework for quantisation across quantum groups and punctured surfaces.

Abstract

We prove that the skein categories of Walker--Johnson-Freyd satisfy excision. This allows us to conclude that skein categories are $k$-linear factorisation homology and taking the free cocompletion of skein categories recovers locally finitely presentable factorisation homology. An application of this is that the skein algebra of a punctured surface related to any quantum group with generic parameter gives a quantisation of the associated character variety.

Excision of Skein Categories and Factorisation Homology

TL;DR

The paper proves that Walker–Johnson-Freyd skein categories satisfy excision, enabling a description of skein categories as k-linear factorisation homology: Sk_V(Σ) ≃ ∫_Σ^{Cat_k^×} 𝒱. It then relates Tambara’s relative tensor product for k-linear categories to the bar-construction approach, showing the two notions of relative tensor product coincide in Cat_k^× and establishing excision for skein categories in the factorisation-homology sense. Building on this, the authors connect skein categories to the LFP_k framework, demonstrate that free cocompletions of skein categories realize LFP_k factorisation homologies, and apply these results to quantised character varieties, proving that skein algebras for generic q yield deformation quantisations of character varieties. The work unifies skein theory with factorisation-homology, providing a robust method to quantise character varieties via skein-theoretic and categorical constructions. The results generalise known cases (e.g., SL_n character varieties) and offer a broad framework for quantisation across quantum groups and punctured surfaces.

Abstract

We prove that the skein categories of Walker--Johnson-Freyd satisfy excision. This allows us to conclude that skein categories are -linear factorisation homology and taking the free cocompletion of skein categories recovers locally finitely presentable factorisation homology. An application of this is that the skein algebra of a punctured surface related to any quantum group with generic parameter gives a quantisation of the associated character variety.

Paper Structure

This paper contains 25 sections, 23 theorems, 87 equations, 14 figures.

Table of Contents

  1. Skein Categories
  2. Definition of a Skein Category
  3. Diagrams
  4. Tambara Relative Tensor Product of $k$-linear Categories
  5. Module Structure and the Relative Tensor Product of Skein Categories
  6. Proof of the Excision of Skein Categories
  7. Equivalence of a non-crossing morphism
  8. Equivalence of middle region
  9. Crossings commute with disjoint morphisms
  10. Skein Categories as $k$-linear Factorisation Homology
  11. The Category Catktimes
  12. Factorisation Homology
  13. Definition
  14. Excision
  15. Characterisation of the Factorisation Homology of Surfaces
  16. ...and 10 more sections

Key Result

Theorem 1

Let $C$ be a $1$-manifold with a thickened right embedding into the boundary of the surface $M$ and a thickened left embedding into the boundary of the surface $N$. There is an equivalence of categories

Figures (14)

  • Figure 1: A ribbon graph.
  • Figure 2: A coloured ribbon graph.
  • Figure 3: The twist $\theta_V: V \to V$ and the braiding $B_{U, V}: U \otimes V \to V \otimes U$ of ${\normalfont\textbf{Ribbon}}_{\mathscr{V}}(C \times [0,1])$.
  • Figure 4: This embedding of surfaces induces a functor ${\normalfont\textbf{Sk}}(M) \times {\normalfont\textbf{Sk}}(N) \to {\normalfont\textbf{Sk}}(M \sqcup_A N)$ of their skein categories. The functor $F$ on $P\left( {\normalfont\textbf{Sk}}(M) \times {\normalfont\textbf{Sk}}(N) \right)$ is given by this functor: that is on objects and on morphisms of the form $(f, g)$.
  • Figure 5: The functor $F$ on the natural isomorphism $\iota$ gives a ribbon which has stands which cross the middle section from $F(\emptyset \lhd a, \emptyset)$ to $F(\emptyset, a \rhd \emptyset)$ (coloured red). Elsewhere applying $F(\iota_{m, a, n})$ only moves points a little.
  • ...and 9 more figures

Theorems & Definitions (99)

  • Theorem 1
  • Theorem 2
  • Theorem 3
  • Corollary 4
  • Definition 1.1
  • Definition 1.2
  • Definition 1.3
  • Definition 1.4
  • Definition 1.5
  • Remark 1.6
  • ...and 89 more