GABRIELLA BÖHM (Hungarian Academy of Sciences, Budapest):
A CATEGORICAL APPROACH TO CYCLIC DUALITY
We construct a large class of para-(co)cyclic objects from 2-functors of a
canonically chosen domain. Their cyclic duality is shown to be governed by
a functor between 2-functor categories. Examples of concrete realizations
of our construction are provided by various para-(co)cyclic modules
arising from Hopf-cyclic theory. (This is joint work with Dragoș
Ștefan.)
GABRIELLA BÖHM (Hungarian Academy of Sciences, Budapest):
WEAK HOPF MONADS
A bialgebra over a field can be characterized as an algebra whose module
category is monoidal with a strict monoidal forgetful functor to the
category of vector spaces. By analogy, in the literature, a bimonad is
defined as a monad on a monoidal category whose Eilenberg-Moore category
of algebras is monoidal with a strict monoidal forgetful functor to the
base category. More generally, we study weak bimonads, i.e., monads on
monoidal categories, with a monoidal Eilenberg-Moore category, but
requiring the forgetful functor to possess only compatible (non-strict)
monoidal and op-monoidal structures. While for a bimonad the monoidal
structure of the Eilenberg-Moore category is lifted from the base
category, in the weak case it is obtained as a canonical retract. A weak
bimonad on a Cauchy complete monoidal category is shown to be equivalent
to a bimonad on another appropriately chosen monoidal category. A weak
bimonad is said to be a weak Hopf monad provided that a canonical (`Galois
type') natural transformation is an isomorphism. Examples of weak bimonads
and weak Hopf monads are provided by weak bialgebras and weak Hopf
algebras, respectively, in braided monoidal categories. (This is joint
work in progress with Stephen Lack and Ross Street.)