Impredicative Encodings of (Higher) Inductive Types

Postulating an impredicative universe in dependent type theory allows System F style encodings of finitary inductive types, but these fail to satisfy the relevant {\eta}-equalities and consequently do not admit dependent eliminators. To recover {\eta} and dependent elimination, we present a method to construct refinements of these impredicative encodings, using ideas from homotopy type theory. We then extend our method to construct impredicative encodings of some higher inductive types, such as 1-truncation and the unit circle S1

Master index

Pla general, del mural ceràmic que decora una de les parets del vestíbul de la Facultat de Química de la UB. El mural representa diversos símbols relacionats amb la química

Polynat in PER-models

The polymorphic lambda-calculus can be modelled using PERs on a partial combinatory algebra. We say that the type of natural numbers (polynat) is polymorphically standard in such a model if the interpretation of the type only contains (the interpretations of) the Church numerals. We show that this is not always the case by constructing an explicit counterexample. On the other hand, when the PCA has either (strong) equality or weak equality plus a form of continuity, we show polynat is standard