I will discuss the idea of writing the semantics of various different languages in the common language of categorical logic. By taking this approach, results for the different languages, such as congruence of bisimilarity, can be established once and for all, in a general setting.

The semantics of a simple process calculus, such as CCS, arises in categorical logic in the category of sets. Different versions of the pi-calculus arise by interpreting categorical logic in more elaborate categories. I am currently investigating still more elaborate settings that accommodate applied pi calculi and higher order calculi.

This field of research involves developing new model theories for different language idioms. The research also sheds new light on work on rule formats, like GSOS and tyft/tyxt. (For those who know the “bialgebraic” work of Turi and Plotkin: there are some connections with this, as well.)