In order to represent, compute and reason with advanced data types one must go beyond the traditional treatment of data types as being inductive types and, instead, consider them as inductive families. Strictly positive types (SPTs) form a grammar for defining inductive types and, consequently, a fundamental question in the the theory of inductive families is what constitutes a corresponding gr...

We introduce container functors as a representation of data types providing a new conceptual analysis of data structures and polymorphic functions. Our development exploits Type Theory as a convenient way to define constructions within locally cartesian closed categories. We show that container morphisms can be full and faithfully interpreted as polymorphic functions (i.e. natural transformatio...