Computer Science Department
School of Computer Science, Carnegie Mellon University


Higher Inductive Types as Homotopy-Initial Algebras

Kristina Sojakova

January 2014

Superceded by CMU-CS-14-1014R


Keywords: Homotopy Type Theory, higher inductive types, homotopy-initial algebras

Homotopy Type Theory is a new field of mathematics based on the surprising and elegant correspondence between Martin-Löf's constructive type theory and abstract homotopy theory. We have a powerful interplay between these disciplines - we can use geometric intuition to formulate new concepts in type theory and, conversely, use type-theoretic machinery to verify and often simplify existing mathematical proofs. A crucial ingredient in this new system are higher inductive types, which allow us to represent objects such as spheres, tori, pushouts, and quotients. We investigate a variant of higher inductive types whose computational behavior is determined up to a higher path. We show that in this setting, higher inductive types are characterized by the universal property of being a homotopy-initial algebra.

51 pages

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