Monoidal 2-categories from foam evaluation

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.

💡 Research Summary

The paper develops a general, rigorous framework for constructing locally linear, semistrict monoidal 2‑categories from a closed foam evaluation formula. The authors begin by observing that many link homology theories—Khovanov homology, Khovanov‑Rozansky gl(N) homology, and their refinements—are naturally expressed in terms of webs (planar graphs) and foams (singular surfaces interpolating between webs). While these objects have been treated either combinatorially (as generators and relations in a 1‑category) or topologically (as embedded surfaces with gluing), a fully fledged higher‑categorical structure has been missing. This work fills that gap.

The input data consist of:

1. A combinatorial type of foam (e.g., oriented facets meeting cyclically at seams, labelled by integers satisfying a flow condition).
2. A closed foam evaluation map taking any closed foam to an element of a commutative ground ring k, satisfying multiplicativity and, in the graded case, compatibility with degree shifts.
3. A rule for admissible embeddings of such foams into oriented 3‑manifolds, ensuring that the cyclic ordering of facets matches the ambient orientation.

From these ingredients the authors first construct a symmetric monoidal k‑linear category C PreFoam of “pre‑foams” between closed labelled webs. The evaluation map induces a bilinear pairing on morphism spaces; its radical defines an ideal of foam relations. Quotienting by this ideal yields the genuine foam category C Foam, which already carries a TQFT‑like functor F: C PreFoam → k‑mod assigning to each closed web its state space.

The second stage upgrades C Foam to a semistrict monoidal 2‑category C Foam (the same notation is used for the 2‑category). Objects are finite collections of labelled points, 1‑morphisms are webs (oriented graphs) between them, and 2‑morphisms are the k‑modules Hom(S,T) defined as F applied to the abstract boundary web abs Hom(S,T). Vertical composition is given by gluing foams along matching boundaries; horizontal composition corresponds to juxtaposing webs and gluing the associated foams side‑by‑side; the monoidal product is disjoint union. The multiplicativity of the closed foam evaluation guarantees associativity and unit laws up to coherent isomorphism, making the structure semistrict.

A major achievement is the proof that every object is 2‑dualizable (has left and right duals) and every 1‑morphism admits left and right adjoints. Moreover, the authors construct a canonical “spatial 2‑duality” (Barrett‑Meusburger‑Schaumann) which encodes oriented 3‑dimensional pivotality and sphericality. In the graded setting, the degrees of the unit and counit of the underlying Frobenius algebras become the grading shifts for adjoints, ensuring that the 2‑category is locally graded‑linear.

Two families of examples instantiate the general theory:

• Bar‑Natan monoidal 2‑categories (BN_A): For any commutative ring k and any commutative Frobenius algebra A free over k, the decorated cobordism construction of Bar‑Natan extends to a locally k‑linear semistrict monoidal 2‑category BN_A generated by a self‑dual object. When k is graded and A is a graded Frobenius algebra, BN_A acquires a graded structure.

• gl(N) foam monoidal 2‑categories (N_Foam): For any commutative ring k and integer N≥1, the Robert‑Wagner closed foam evaluation for gl(N) foams yields a locally graded‑linear semistrict monoidal 2‑category N_Foam. Objects are labelled by the set {1,…,N}, 1‑morphisms are gl(N) webs, and 2‑morphisms are generated by gl(N) foams. All objects have duals, all 1‑morphisms have adjoints, and the spatial duality structure is present.

The paper emphasizes that these 2‑categories provide the algebraic backbone for extending link homology theories from knots to tangles: tangles become 1‑morphisms, cobordisms between tangles become 2‑morphisms, and the chain complexes appearing in link homology arise from applying the TQFT to the foam relations. The authors also discuss how, after passing to idempotent completion and taking bounded homotopy categories, one obtains stable, locally linear (∞,2)‑categories that are expected to carry E₂‑monoidal structures, potentially serving as fully local 4‑dimensional TQFTs.

Finally, the authors compare their construction with existing notions of rigidity, pivotality, and sphericality in monoidal 1‑categories, showing that the higher‑categorical analogues hold in their examples. They note that while the graded version yields genuine duality structures, the ungraded version with only a ℤ‑action on hom‑categories loses the precise dualities because left and right adjoints differ by non‑trivial grading shifts.

In summary, the paper delivers a clean, combinatorial route from a closed foam evaluation to a richly structured monoidal 2‑category, validates the construction on two central families of foams relevant to modern link homology, and sets the stage for further developments in higher‑dimensional topological quantum field theory.