Monoidal 2-categories from foam evaluation

TL;DR

This paper develops a framework for constructing monoidal 2-categories from foam evaluations, connecting link homology theories with categorical structures, and providing explicit examples related to Khovanov and general linear link homologies.

Contribution

It introduces a general method for building monoidal 2-categories from foam evaluations, with concrete examples linked to well-known link homology theories.

Findings

Constructed monoidal 2-categories based on gl(N)-foams and Bar-Natan's decorated cobordisms.

These categories are non-semisimple, with duals, adjoints, and a spatial duality structure.

The framework unifies various link homology theories within a categorical context.

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.