Frobenius algebras and ambidextrous adjunctions

TL;DR

This paper explores the deep relationship between Frobenius objects in monoidal categories and ambidextrous adjunctions in 2-categories, extending the theory to higher categories and linking it to 2D topological quantum field theories.

Contribution

It establishes a correspondence between Frobenius objects and ambijunctions in 2-categories, and extends this to Frobenius pseudomonoids and pseudo ambijunctions in higher categories.

Findings

Every Frobenius object arises from an ambijunction in some 2-category.

Every Frobenius pseudomonoid arises from a pseudo ambijunction in some 3-category.

The results connect algebraic structures with topological quantum field theories.

Abstract

In this paper we explain the relationship between Frobenius objects in monoidal categories and adjunctions in 2-categories. In particular, we show that every Frobenius object in a monoidal category M arises from an ambijunction (simultaneous left and right adjoints) in some 2-category D into which M fully and faithfully embeds. Since a 2D topological quantum field theory is equivalent to a commutative Frobenius algebra, this result also shows that every 2D TQFT is obtained from an ambijunction in some 2-category. Our theorem is proved by extending the theory of adjoint monads to the context of an arbitrary 2-category and utilizing the free completion under Eilenberg-Moore objects. We then categorify this theorem by replacing the monoidal category M with a semistrict monoidal 2-category M, and replacing the 2-category D into which it embeds by a semistrict 3-category. To state this more powerful result, we must first define the notion of a `Frobenius pseudomonoid', which categorifies that of a Frobenius object. We then define the notion of a `pseudo ambijunction', categorifying that of an ambijunction. In each case, the idea is that all the usual axioms now hold only up to coherent isomorphism. Finally, we show that every Frobenius pseudomonoid in a semistrict monoidal 2-category arises from a pseudo ambijunction in some semistrict 3-category.