3
$\begingroup$

Starting with a monoidal category with duals $C$, one may consider the category $End(C)$ of endofunctors of $C$. A Hopf monad on $C$ is a bimonad on $C$ with (a generalised notion of the) antipode. Under suitable assumptions for the Hopf monad, this Hopf monad applied on the unit object of the category $C$ is a Hopf algebra in the centre of $C$.

The question is if the notion of Poisson monad in $End(C)$ has been defined somewhere? In the case that such a notion exists, does it return a Poisson algebra object in the category?

Improve this question
$\endgroup$
4
  • 1
    $\begingroup$ How does the monoidal structure with duals relate to your question? $\endgroup$
    – მამუკა ჯიბლაძე
    Commented Jan 12, 2021 at 16:02
  • 1
    $\begingroup$ @მამუკაჯიბლაძე The category of interest is monoidal with duals, and is essential for the Hopf monad case. Probably for the Poisson is not needed. $\endgroup$
    – Christos
    Commented Jan 12, 2021 at 16:48
  • $\begingroup$ What is your reference for Hopf monads? You see, there is a notion by Mesablishvili & Wisbauer which does not use any structure on the category except idempotent splitting, and there are others using monoidal structure essentially; if you mean the latter, there are several versions of these, which one do you mean? As for the Poisson structure, I believe you cannot formulate it without some form of additivity requirements, both on your category and on the endofunctor. $\endgroup$
    – მამუკა ჯიბლაძე
    Commented Jan 12, 2021 at 17:39
  • $\begingroup$ @მამუკაჯიბლაძე My reference for Hopf monads is that of Brugieres-Virelizier. $\endgroup$
    – Christos
    Commented Jan 12, 2021 at 21:16

0

Reset to default

You must log in to answer this question.

Browse other questions tagged .