Formal Hopf algebra theory, II: Lax centres

This paper is the second in a series started by [Ignacio L. López Franco, Formal Hopf algebra theory I: Hopf modules for pseudomonoids, J. Pure Appl. Algebra 213 (2009) 1046–1063], aiming to extend the basic theory of Hopf algebras to the context of pseudomonoids in monoidal bicategories. This article concentrates on the notion of lax centre of a pseudomonoid and its relationship with the Drinfel’d or quantum double of a finite Hopf algebra and the centre of a monoidal category. We can distinguish two parts in the present paper. In the first, for a pseudomonoid AA with lax centre ZℓAZℓA in a Gray monoid ℳℳ with certain extra properties, we exhibit an equivalence ℳ(I,ZℓA)≃Zℓ(ℳ(I,A))ℳ(I,ZℓA)≃Zℓ(ℳ(I,A)) of categories enriched in ℳ(I,I)ℳ(I,I). In the second, we construct the lax centre of a left autonomous map pseudomonoid AA as an Eilenberg–Moore object for a certain opmonoidal monad on AA. Moreover, if AA is also right autonomous, the lax centre coincides with the centre. As an application, we show that a (left) autonomous monoidal VV-category has a (lax) centre in V-Mod, of which we give an explicit description. In another application, we prove that a finite-dimensional coquasi-Hopf algebra HH has a centre in the monoidal bicategory Comod(Vect) and it is equivalent to the Drinfel’d double D(H)D(H).