University of Birmingham > Talks@bham > Lab Lunch > Commutative or monoidal monads

Commutative or monoidal monads

Add to your list(s) Download to your calendar using vCal

If you have a question about this talk, please contact Dr Steve Vickers.

The purpose of this talk is to ask the audience’s help in understanding the title.

Steve Vickers and I are working towards the notion of colocale, but coming to it from completely different directions. My approach is from pure category theory, asking what interesting things you can say about adjunctions that are both monadic and comonadic.

Limits of algebras are easy, as are colimits of coalgebras, but after making both steps you’re lost.

However, the category of coalgebras over algebras has binary products so long as the original category has them and the monad has a strength.

This by-passes consideration of the algebras alone, but in the spirit of treating both ends of the adjunction with equal importance, we should find out what is going on there.

Unwinding my construction of products of coalgebras, we get tensor products of algebras, but only under additional hypotheses.

The notion of strength of a monad is manifestly asymmetrical. Rectifying this gives a commutative monad.

In the concrete direction, I think that commutative monads correspond to algebraic theories that are commutative in the sense that every operation symbol commutes with every other and so a homomoprhism.

More abstractly, a commutative monad is apparently the same as a monadic monad.

This construction yields the familiar tensor products of vector spaces and of complete semilattices, so the ideas are familiar in several disciplines.

This talk is part of the Lab Lunch series.

Tell a friend about this talk:

This talk is included in these lists:

Note that ex-directory lists are not shown.

 

Talks@bham, University of Birmingham. Contact Us | Help and Documentation | Privacy and Publicity.
talks@bham is based on talks.cam from the University of Cambridge.