Autonomous Pseudomonoids

Abstract

In this dissertation we generalise the basic theory of Hopf algebras to the context of autonomous pseudomonoids in monoidal bicategories. Autonomous pseudomonoids were introduced in [13] as generalisations of both autonomous monoidal categories and Hopf algebras. Much of the theory of autonomous pseudomonoids developed in [13] was inspired by the example of autonomous (pro)monoidal enriched categories. The present thesis aims to further develop the theory with results inspired by Hopf algebra theory instead. We study three important results in Hopf algebra theory: the so-called "fundamental theorem of Hopf modules", the "Drinfel'd quantum double" and its relation with the centre of monoidal categories, and " Radford's formula". The basic result of this work is a general fundamental theorem of Hopf modules that establishes conditions equivalent to the existence of a left dualization. With this result as a base, we are able to construct the centre (defined in [83]) and the lax centre of an autonomous pseudomonoid as an Eilenberg-Moore construction for certain monad. As an application we show that the Drinfel'd double of a finite-dimensional Hopf algebra is equivalent to the centre of the associated pseudomonoid. The next piece of theory we develop is a general Radford's formula for autonomous map pseudomonoids formula in the case of a (coquasi) Hopf algebra. We also introduce "unimodular" autonomous pseudomonoids. In the last part of the dissertation we apply the general theory to enriched categories with a (chosen) class of (co)limits, with emphasis in the case of finite (co)limits. We construct tensor products of such categories by means of pseudo-commutative enriched monads (a slight generalisation of the pseudo-commutative 2-monads of [37], and showing that lax-idempotent 2-monads are pseudo-commutative. Finally we apply the general theory developed for pseudomonoids to deduce the main results of [27].This work was supported by an Internal Graduate Studentship, Trinity College, Cambridge