Closed Reconstruction

Abstract: Starting with a commutative ring R, it is well-known that monoidal structures on the category of H-modules, for an R-algebra H, that are compatible with the canonical forgetful functor are in bijection with bialgebra structures on H. If H is even Hopf, then it is also possible to lift the closed structure from R-mod to H-mod.

In some cases one might be interested in first lifting just the closed structure, without any assumption of monoidal compatibility. This talk will focus on reconstruction of this kind, more closely studying the question from the monadic point of view. While in the monoidal case, oplax monoidal monads are the right notion, in the closed world, (lax) closed monads are inadequate for reconstruction purposes. The right notion turns out to be that of a gabi-monad, first outlined by Berger, Saracco, and Vercruysse. In the algebraic setting, lifting the closed structure in this way already implies that one started with a Hopf algebra. This, however, is not the case for monads: we give examples of gabi-monads that are not Hopf.

The talk is based on joint work in progress with Sebastian Halbig and Paolo Saracco.

Mathematics

Audience: researchers in the topic

European Quantum Algebra Lectures (EQuAL)

Series comments: EQuAL is an online seminar series on quantum algebra and related topics such as topological and conformal field theory, operator algebra, representation theory, quantum topology, etc. sites.google.com/view/equalseminar/home