Impredicative Encodings of (Higher) Inductive Types

Steve Awodey; Jonas Frey; Sam Speight

doi:10.1145/3209108.3209130

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arXiv · 1802.02820

Impredicative Encodings of (Higher) Inductive Types

Steve Awodey·Jonas Frey·Sam Speight

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Abstract

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.

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Steve Awodey, Jonas Frey, Sam Speight. 2018-02-08. Impredicative Encodings of (Higher) Inductive Types. https://doi.org/10.1145/3209108.3209130

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