Lecture Notes on Cubical sets

These are some lecture notes for a course presenting the cubical set model of type theory, first in Copenhagen, December 2014, and then in Paris, February 2015. We describe a particular presheaf model of type theory. This description can also be seen as an operational semantics of a purely syntactical type system. It involves a nominal extension of λ-calculus. We use a generalization of the Kan composition operation which is represented by a nominal operation. Any element represents a hypercube, the direction being represented by the symbols this element depends on. One can define the notion of homotopy between two cubes (it is a cube with one new dimension connecting the two given cubes). The operation we introduce consists in changing some faces along given homotopies; Kan composition is the special case where we replace all faces. One new feature, compared with usual λ-calculus or first-order logic, is how this operation interacts with nominal substitution.