Homotopy Type Theory For Dummies

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Metatheory Canonicity We have introduced additional constants proving equalities: ◮ functional extensionality ◮ univalence ◮ equality constructors for HITs While we have introduced some definitional equality, they are insufficient to guarantuee canonicity: All closed terms of type N are definitionally equal (indeed reducible) to a numeral. While we (with Conor McBride and Wouter Swierstra) have developed an approach to solve the problem for functional extensionality (Observational Type Theory), this relies on a strong form of UIP. Thorsten Altenkirch (Nottingham) Edinburgh 13 October 30, 2013 26 / 29
Metatheory Constructive models The homotopy interpretation of type theory uses Kan Fibrations in simplicial sets (the simplicial set model). This construction relies essentially on classical principles. Thierry Coquand has suggested a modification of this construction using semi-simplicial sets. This models only weak Type Theory (i.e. no conversion under λ). An alternative would be to directly use weak ω-groupoids. It seems likely that an answer to this question would also provide a solution to the canonicity problem. Thorsten Altenkirch (Nottingham) Edinburgh 13 October 30, 2013 27 / 29
Page 1 and 2: Homotopy Type Theory For Dummies Th
Page 3 and 4: Basic Type Theory Type Theory 101 M
Page 5 and 6: Identity types Identity types Given
Page 7 and 8: Identity types Should equality be p
Page 9 and 10: Homotopic interpretation Uniqueness
Page 11 and 12: Homotopic interpretation While we c
Page 13 and 14: The univalence axiom Univalence Rej
Page 15 and 16: The univalence axiom What is the eq
Page 17 and 18: Truncation levels Truncation levels
Page 19 and 20: Truncation levels The Truncation Hi
Page 21 and 22: HITs Higher inductive types There a
Page 23 and 24: HITs Higher inductive types This al
Page 25: HITs Truncations We can use HITs to
Page 29: Computer Science ? The HoTT people

Homotopy Type Theory For Dummies

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