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Higher Inductive Types as Homotopy-Initial Algebras

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Technical Report

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Carnegie Mellon University Pittsburgh United States

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Homotopy type theory is a new field of mathematics based on the recently discovered correspondence between Martin-Lofs 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 from a variety of domains, such as spheres, tori, pushouts, and quotients, in the type theory. In this thesis we formulated and investigated a class of higher inductive types we call W-quotients which generalize Martin-Lofs well-founded trees to a higher dimensional setting. We have shown that a propositional variant of W-quotients, 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 derived that W-quotients in the strict form are homotopy-initial. A number of other interesting higher inductive types such as spheres, suspensions, and type quotients arise as special cases of W-quotients, and the characterization of these as homotopy initial algebras thus follows as a corollary to our main result. We have investigated further higher inductive types - arbitrary truncations, set quotients, and groupoid quotients - and established analogous characterizations interms of the universal property of homotopy-initiality. Furthermore, we showed that set and groupoid quotients can be recovered from W-quotients and truncations.

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  • Numerical Mathematics

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