Skein Categories in Non-semisimple Settings

TL;DR

The paper develops a non-semisimple extension of skein theory by introducing skein categories with respect to a tensor ideal $\\mathcal{I}$ in a ribbon category, and shows these modified skein categories compute factorization homology for the corresponding disk algebra. The main result proves an exact equivalence between factorization homology with coefficients in $\\mathcal{I}$ and the $\\mathcal{I}$-skein category, implemented via the distinguished presheaf and endomorphism skein algebras. This framework yields a robust 2D description compatible with a higher-dimensional non-semisimple Crane--Yetter TQFT, describes new elements arising from bichrome graphs, and provides concrete computations of $\\mathcal{I}$-skein algebras via Lyubashenko’s coend. The work thus integrates topological skein constructions with the modern factorization-homology formalism to extend extended TQFTs to non-semisimple settings, with potential applications to non-semisimple 3- and 4-manifold theories.

Abstract

We introduce a version of skein categories of surfaces which depends on a tensor ideal in a linear ribbon category, thereby extending the existing theory to the setting of non-semisimple TQFTs. We obtain modified notions of skein algebras of surfaces and skein modules of 3-cobordisms for non-semisimple ribbon categories. We prove that these skein categories built from ideals coincide with factorization homology, shedding new light on the similarities and differences between the semisimple and non-semisimple settings. The essential difference is the need to work with profunctors in the non-semisimple setting. Doing so produces a ``distinguished presheaf'' which plays the role of the distinguished object in skein categories in semisimple settings. As a consequence, we get a skein-theoretic description of factorization homology for a large class of balanced braided presentable categories, precisely all those which are expected to induce oriented categorified 3-TQFTs.

Figures (5)

• Figure 1: Left: an $\mathcal{I}$-labeling on the disjoint union of two disks. It is not admissible because the lower disk is unlabeled. Right: a compatible $\mathcal{I}$-ribbon graph in the thickened disk. The chosen normal on the two disks is going up on the picture. It is admissible.
• Figure 2: A mixed vertex in a bichrome graph. It is colored by a morphism $f: {\mathrm{1}\mkern-4mu{}\mathrm{l}} \to \mathcal{L}\otimes \mathcal{L} \otimes P^*$ in $\widehat{\mathcal{I}}$, i.e. a natural transformation from $\mathop{\mathrm{Hom}}\nolimits_\mathcal{A}(-,{\mathrm{1}\mkern-4mu{}\mathrm{l}})$ to $\int^{X,Y\in\mathcal{I}}\mathop{\mathrm{Hom}}\nolimits(-,X\otimes X^*\otimes Y \otimes Y^* \otimes P^*)$.
• Figure 3: The red-to-blue operation. Here $\tilde{f} : P \to X \otimes X^*\otimes Y \otimes Y^*$ is a representative of $f: P \to \mathcal{L} \otimes \mathcal{L}$, and $P:=Q\otimes Q^*$.
• Figure 4: The natural transformation $\tilde{E} \overset{c}{\Rightarrow} \mathop{\mathrm{const}}\nolimits_{\int_{\Sigma'}\!E}\overset{\mathop{\mathrm{const}}\nolimits_G}{\Longrightarrow}\mathop{\mathrm{const}}\nolimits_X$, with all 2-morphisms suppressed.
• Figure 5:

Theorems & Definitions (55)

• Definition 2.1
• Definition 2.2
• Definition 2.3
• Remark 2.4
• Definition 2.5
• Remark 2.6
• Definition 2.7
• Definition 2.8
• Theorem 2.9
• Remark 2.10