Homotopy Type Theory For Dummies

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The univalence axiom What is the equality of types? We can show: coe : A = B → A ≈ B The univalence axiom sates that coe is an equivalence. This is justified by a black box view of types. Note that the univalence axiom implies functional extensionality. Thorsten Altenkirch (Nottingham) Edinburgh 13 October 30, 2013 16 / 29
Truncation levels Truncation levels In the absence of UIP we classify types according to the complexity of their equality types. We say a type A : Type is contractible or a -2-type, if is inhabited. Σ a:A Π x:A x = a A type A : Type is a n + 1-type, if all its equality types a = A a ′ for a, a ′ : A an n-types. To show that any n-Type is also a n + 1-type we need to show that if a type is contractible, then its equalities are contractible too. Thorsten Altenkirch (Nottingham) Edinburgh 13 October 30, 2013 17 / 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: The univalence axiom What is the eq
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 and 26: HITs Truncations We can use HITs to
Page 27 and 28: Metatheory Constructive models The
Page 29: Computer Science ? The HoTT people

Homotopy Type Theory For Dummies

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