Computer Science Department
School of Computer Science, Carnegie Mellon University


Higher Inductive Types as Homotopy-Initial Algebras

Kristina Sojakova

January 2014
Revised July 2014

This report is a revised version of CMU-CS-14-101


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

Homotopy Type Theory is a new field of mathematics based on the recently-discovered 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. Higher inductive types form a crucial part of this new system since they allow us to represent mathematical objects, such as spheres, tori, pushouts, and quotients, in the type theory. We investigate a class of higher inductive types called W-suspensions which generalize Martin-Löf's well-founded trees. We show that a propositional variant of W-suspensions, whose computational behavior is determined up to a higher path, is characterized by the universal property of being a homotopy-initial algebra. As a corollary we get that W0-suspensions in the strict form are homotopy-initial.

51 pages

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