In recent years, theories that combine algebraic and coalgebraic operations have been increasingly prevalent in computer science, mathematics and physics. These include graphical approaches to the study of quantum information of Abramsky, Coecke, Duncan and others, calculi of connectors of Bruni, Lanese and Montanari for combining software components and Rutten's calculi of stream circuits, amongst others. Product and permutation categories (PROPs) provide a convenient mathematical framework that allows the rigorous study of such theories. PROPs come with a toolbox of constructions, for example distributed laws, which lead to modular decomposition of theories and lead to characterisations of free models. I will present the basic theory of PROPs, and give several examples.
This is joint work with Filippo Bonchi and Fabio Zanasi.