Coherence for Frobenius pseudomonoids and the geometry of linear proofs

Frobenius pseudomonoids are higher-dimensional algebraic structures, first studied by Street [34], which categorify the classical algebraic notion of Frobenius algebra [24]. These higher algebraic structures have an important application to logic, since Frobenius pseudomonoids in the bicategory of categories, profunctors and natural transformations, for which the multiplication and unit have right adjoints, correspond to ∗-autonomous categories [4, 5], the standard categorical semantics for multiplicative linear logic. They also play a central role in topological quantum field theory [8, 9, 24, 35]. Our main result is a coherence theorem for Frobenius pseudomonoids. In the second part of the paper, we apply this coherence theorem to the problem of geometrical proof representation in linear logic, giving a 3d notation for proofs with a geometrical notion of equivalence.