2019-10-24: Joe Collins (MSP)

Tue, 14 Oct 2025 16:51:42 +0000

Joe Collins (MSP)

Hopf Monads and Formally Adding in Morphisms

Abstract

It should be noted that this is ongoing work. The talk that you will see is going to be very unpolished; this is stuff that I have been working on for the past while, and the material will not be fully formed yet. The things I say may not end up matching with reality. So with that minor caveat, let us continue.

A Hopf algebra is a monoid, a comonoid, and an extra endomorphism such that various commutation rules are obeyed, and Hopf monads are a strange generalisation of Hopf algebras in the direction of monads. I have previously talked about when a Hopf algebra is a Hopf-Frobenius algebra, and my goal now is to generalise this to the case of Hopf Monads. However, to keep things interesting, I am doing this in a weird way.

Let T be a Hopf monad in monoidal category C. T is isomorphic to a Hopf algebra if there exists a natural transformation e: T -> 1 that respects the Hopf monad structure. There are many examples of Hopf monads which are not isomorphic to any Hopf algebras, so this morphism does not exist in C in general. However, what happens if I formally add in this morphism, creating a new category C_e? In this category, T would presumably be isomorphic to a Hopf algebra, and this begs the question: what can T in C_e tell us about T in C? By looking at the natural functor C -> C_e, can I use this to transfer my theorem about Hopf algebras to Hopf monads? I certainly believe so, but let's find out together!

2019-10-24 13:30:00 UTC
LT1310