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Monoidal 2-categories from foam evaluation

Leon J. Goertz, Laura Marino, Paul Wedrich

TL;DR

The paper introduces a general, algebro-combinatorial framework for building locally linear semistrict monoidal 2-categories from closed foam evaluations, unifying Bar-Natan decorated cobordisms and $\mathfrak{gl}_N$-foam theories. By combining a universal construction for TQFTs of webs and foams with holographic 2-morphism indexing, it produces explicit semistrict monoidal 2-categories, notably $\mathrm{BN}_A$ and $N\mathbf{Foam}$, that admit duals for objects and adjoints for 1-morphisms and carry a canonical spatial duality structure. The work develops graded refinements via Frobenius algebras and shows how the constructions model link homology phenomena on the 2-categorical level, providing a structured setting for extended TQFTs and higher-categorical skein theories. These frameworks pave the way for rigorous categorifications of general linear link homologies and decorated cobordism theories, with potential connections to higher TQFTs and 3D pivotal/spherical dualities in a unified foam-based language.

Abstract

In this paper we describe a general framework for constructing examples of locally linear semistrict monoidal 2-categories covering many examples appearing in link homology theory. The main input datum is a closed foam evaluation formula. As examples, we rigorously construct semistrict monoidal 2-categories based on gl(N)-foams, which underlie the general linear link homology theories, and further examples based on Bar-Natan's decorated cobordisms, related to Khovanov homology. These monoidal 2-categories are typically non-semisimple, have duals for all objects, adjoints for all 1-morphisms, and carry a canonical spatial duality structure expressing oriented 3-dimensional pivotality and sphericality.

Monoidal 2-categories from foam evaluation

TL;DR

The paper introduces a general, algebro-combinatorial framework for building locally linear semistrict monoidal 2-categories from closed foam evaluations, unifying Bar-Natan decorated cobordisms and -foam theories. By combining a universal construction for TQFTs of webs and foams with holographic 2-morphism indexing, it produces explicit semistrict monoidal 2-categories, notably and , that admit duals for objects and adjoints for 1-morphisms and carry a canonical spatial duality structure. The work develops graded refinements via Frobenius algebras and shows how the constructions model link homology phenomena on the 2-categorical level, providing a structured setting for extended TQFTs and higher-categorical skein theories. These frameworks pave the way for rigorous categorifications of general linear link homologies and decorated cobordism theories, with potential connections to higher TQFTs and 3D pivotal/spherical dualities in a unified foam-based language.

Abstract

In this paper we describe a general framework for constructing examples of locally linear semistrict monoidal 2-categories covering many examples appearing in link homology theory. The main input datum is a closed foam evaluation formula. As examples, we rigorously construct semistrict monoidal 2-categories based on gl(N)-foams, which underlie the general linear link homology theories, and further examples based on Bar-Natan's decorated cobordisms, related to Khovanov homology. These monoidal 2-categories are typically non-semisimple, have duals for all objects, adjoints for all 1-morphisms, and carry a canonical spatial duality structure expressing oriented 3-dimensional pivotality and sphericality.
Paper Structure (22 sections, 22 theorems, 170 equations, 2 figures)

This paper contains 22 sections, 22 theorems, 170 equations, 2 figures.

Table of Contents

  1. Introduction
  2. Foams and higher categories
  3. Combinatorial vs. topological foams
  4. Approach
  5. From closed foam evaluation to a TQFT for webs and foams
  6. Via holography to monoidal 2-categories
  7. Closed foam evaluation
  8. Closed foams
  9. Evaluation
  10. TQFT for webs and foams
  11. Foams with boundary and webs
  12. Labeled foams with boundary
  13. Universal construction
  14. Pairing and sesquilinearity
  15. Frobenius algebras and gradings
  16. ...and 7 more sections

Key Result

Proposition 3.10

The following data defines a small symmetric monoidal category, which we will denote by $\mathsf{PreFoam}$ and call the category of closed webs and prefoams:

Figures (2)

  • Figure 1: a) Coordinate system for the graphical calculus, b) An embedded trivalent foam as 2-morphism between parallel webs $S,T\colon s \rightarrow t$. c) The associated boundary web. d) A flattened depiction of the boundary web. Throughout we suppress labels and cyclic orderings.
  • Figure 2: Example shapes of top-dimensional morphisms in a) a disklike 3-category, b) a canopolis BN2, c) a monoidal 2-category, d) a triple category.

Theorems & Definitions (101)

  • Remark : On link homology
  • Remark : On skein theory
  • Remark : Graded $\Bbbk$-linear vs. $\Bbbk$-linear with $\mathbb{Z}$-action
  • Remark : Skeins and skein relations
  • Remark : Alternative 3-categorical formalizations
  • Remark
  • Definition 2.1
  • Remark 2.2
  • Definition 2.3
  • Definition 2.4: Labeling data for trivalent and nilvalent foams
  • ...and 91 more