### Tuesday, July 12, 2016 - 10:00am to 11:30am

### Location:

8102 Gates & Hillman Centers### Speaker:

KRISTINA SOJAKOVA, Ph.D. Student http://www.cs.cmu.edu/~ksojakov/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 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-Löf’s 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 in terms of the universal property of homotopy-initiality. Furthermore, we showed that set and groupoid quotients can be recovered from W-quotients and truncations." Thesis Committee: Frank Pfenning (Co-Chair) Steve Awodey (Co-Chair) Robert Harper Thierry Coquand (University of Gothenburg)Nicola Gambino (University of Leeds)