Algebraic Structures Derived from Foams

Abstract

Foams are surfaces with branch lines at which three sheets merge. They have been used in the categorification of sl(3) quantum knot invariants and also in physics. The 2D-TQFT of surfaces, on the other hand, is classified by means of commutative Frobenius algebras, where saddle points correspond to multiplication and comultiplication. In this paper, we explore algebraic operations that branch lines derive under TQFT. In particular, we investigate Lie bracket and bialgebra structures. Relations to the original Frobenius algebra structures are discussed both algebraically and diagrammatically.

Coherence for braided and symmetric pseudomonoids

“…By categorification of the representation theory of Hopf algebras, bicategories encoding the data of a four-dimensional topological field theory can be obtained as representation categories of certain 'Hopf' pseudomonoids [32,8]. Pseudomonoids have also appeared recently in the the theory of surface foams, where certain pseudomonoids in a braided monoidal category represent knotted foams in four-dimensional space [7]. Many properties of monoidal categories can be formulated externally as structures on pseudomonoids; for instance, Street showed that Frobenius pseudomonoids correspond to star-autonomous categories [36], giving rise to a diagrammatic calculus for linear logic [13].…”