The notion of a species of structures was introduced by André Joyal to provide a combinatorial interpretation of the calculus of formal power series. In my talk, I will present a generalisation of the notion of a species of structures and describe their calculus.
One reason for the interest in generalised species of structures is that their calculus gives rise to a cartesian closed structure, and supports in a natural way an operation of differentiation. The combination of these facts gives rise to a model of Ehrhard and Regnier's differential lambda-calculus. I will also explain how this model is related to the familiar relational model of linear logic.
Joint work with Marcelo Fiore, Martin Hyland, and Glynn Winskel.