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 99

Remarks on Units of Skew Monoidal Categories

“…For the following diagram, the square commutes by the naturality of α and the outside commutes by (3). The semicircle on the left commutes as ϕ is an isomorphism.…”

“…Thus each of the equations (1), (3) and (4) is independent of the remaining four equations. By reversing the tensor, direction of arrows and the order of composition we notice that equation (2) and equation (4) are dual, so statements such as independence holds for one if and only if it holds for the other.…”

“…In this section we denote them with the symbol ∼ = as above. These 2-cells satisfy four axioms which we do not list but will appeal to throughout the rest of this section; see [3] once again. We write I for the unit object of the Gray monoid.…”

Spectra associated to symmetric monoidal bicategories

“…For a precise definition and a historical account of the theory of symmetric monoidal bicategories we refer the reader to Schommer-Pries [24]. Incomplete versions of the definition can be found in McCrudden [19] (definition of sylleptic monoidal bicategory) and Day and Street [5] (definition of symmetric Gray monoid). Shulman [27] provides a way of constructing examples of symmetric monoidal bicategories.…”

“…The definition of symmetric monoidal bicategory is cumbersome. Some incomplete definitions can be found in McCrudden [19] and Day and Street [5], while the most concise and complete definition can be found in Schommer-Pries [24]. Shulman [27] shows how to obtain symmetric monoidal bicategories from symmetric monoidal double categories, which are easier to understand.…”