Monoidal Bicategories and Hopf Algebroids

Abstract

Why are groupoids such special categories? The obvious answer is because all arrows have inverses. Yet this is precisely what is needed mathematically to model symmetry in nature. The relation between the groupoid and the physical object is expressed by an action. The presence of inverses means that actions of a groupoid G behave much better than actions of an arbitrary category. The totality of actions of G on vector spaces forms a category Mod G of modules. The feature of Mod G which epitomises the fact that G is a groupoid is that the internal hom in Mod G is calculated in a particularly simple way. More precisely, the functor out of Mod G which forgets the actions preserves, not only the monoidal structure but also, the closed structure. With this as a guiding principle, we develop a general concept of ``autonomous pseudomonoid'' which includes ordinary Hopf algebras (indeed, Hopf algebroids) and autonomous (=compact=rigid) monoidal categories. This is intended to elucidate the interaction between Hopf algebras and autonomous monoidal categories in Tannaka duality as appearing in [JS2; D2], for example. Given a topological monoid M, it is explained in [JS2, Section 8] why the monoidal category of finite-dimensional representations of M is equivalent to the monoidal category of finite-dimensional comodules over the bialgebra R(M ) of representative functions on M. This provides evidence that, when regarding a Hopf algebra as a quantum group, it is the finitedimensional comodules (rather than modules) which should be regarded as the representations of the group. In dealing with comodules, we are using the coalgebra structure of the Hopf algebra H. A Hopf algebroid is an additive category (that is, ``algebra with several objects''; that is, algebroid) with a comonoidal structure: modules over a Hopf algebroid make sense but comodules are not appropriate. What we need is a notion where the article no. AI971649