Stefaan Caenepeel

We introduce multi monads (or monads with several objects) in a 2-category (or bicategory). 

These are given by a one-cells A_{ab}: b |-> a, where a and b run over a subclass of the class of objects of the 2-category, with appropriate (multiplication and unit) structure maps.

A monad with one object turns out to be a monad in the classical sense. Enriched categories appear as examples, and also Morita contexts are examples. 

Modules over multi monads can be introduced. Dually, one can introduce multi comonads and comodules. 

Multi Frobenius monads are at the same time multi monads and comonads, with appropriate compatibility conditions. 

We generalise some classical properties: modules over a Frobenius multi monad are equivalent to comodules; multi Frobenius monads have a self dual property.