Composing theories using distributive laws
Andrei Akhvlediani (Oxford)
Wednesday 26th May, 2010 16:00-17:00 204
Lawvere theories are categories that encode the signature of algebraic structures. It was recently noted that certain constructions in theoretical computer science can be eﬃciently described using the language of Lawvere theories. To combine those constructions, one needs a universal way of combining the associated Lawvere theories. PROPs are the symmetric monoidal analogue of Lawvere theories: they describe algebraic signatures of structures that live in symmetric monoidal categories. The ability to combine simple PROPs to form more complicated ones proved to be instrumental in categorical semantics of quantum informatic protocols. In this talk we shall present a single method for combining Lawvere theories and PROPs. We shall see that both of those structures can be seen as monads in a certain bicategory and that this point of view allows us to combine them using so-called distributive laws. To illustrate this idea, we will present the PROPs for bialgebras and Frobenius algebras as composite PROPs. We shall ﬁnish with some applications to theoretical computer science and quantum information.