[Joint work with Brian Day]
[A little open problem about monoidal adjunctions was presented to start.] Last week (at the University of Sydney) I discussed the contravariant adjunction between V-opcategories and V-categories: each V-category C yields a V-opcategory C^(czech) and each V-opcategory A yields a V-category Comod_{f} A.
This week I looked at refining this when C is monoidal and A is comonoidal (a one-object comonoidal V-opcategory is a bialgebra). This is further refined when C is autonomous (=3D rigid) and A is a “Hopf opalgebroid”. The ideas of my old generalised Beck monadicity theorem from the paper Two constructions on lax functors, Cahiers topologie et geometrie differentielle 13 (1972) 217-264 are applied.