Homotopy Type Theory For Dummies

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Truncation levels Higher types How do we get types at higher levels? The 1st universe <strong>Type</strong> 0 cannot be a set. Consider Bool = Bool, using univalence there are two equivalences: id, and negation. However, id and negation cannot be equal. My student Nicolai Kraus proved that the nth universe is an cannot be an n-type. Thorsten Altenkirch (Nottingham) Edinburgh 13 October 30, 2013 20 / 29
HITs Higher inductive types There are other ways to obtain higher types in HoTT. A higher inductive type has not only constructors for elements but also for equalities. A simple example is S 1 : <strong>Type</strong> base : S 1 loop : base = base S 1 homotopically behaves like the circle. Indeed we can embedd i : Z into φ(i) : base = base: i ↦→loop n 0 ↦→refl − n ↦→(loop −1 ) n Thorsten Altenkirch (Nottingham) Edinburgh 13 October 30, 2013 21 / 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: Truncation levels The Truncation Hi
Page 23 and 24: HITs Higher inductive types This al
Page 25 and 26: HITs Truncations We can use HITs to
Page 27 and 28: Metatheory Constructive models The
Page 29: Computer Science ? The HoTT people

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