Homotopy Type Theory For Dummies

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The univalence axiom What is equality of functions ? Given f , g : A → B we define extensional equality : f ∼ g :≡ Π a:A f (a) = g(a) We can show app : f = g → f ∼ g Extensionality corresponds to requiring app to be an isomorphism. It’s inverse is functional extensionality. This can be justified by the black box view of functions. Thorsten Altenkirch (Nottingham) Edinburgh 13 October 30, 2013 14 / 29
The univalence axiom What is the equality of types? Given A, B : <strong>Type</strong> we define isomorphism A ∼ B as f : A → B g : B → A η : Π a:A a = g(f (a)) ɛ : Π b:B b = f (g(b)) In the absence of UIP we can require coherence properties which correspond to the triangle equalities of an adjunction: δ : Π a:A ap(f , η(a)) = ɛ(f (b)) δ ′ : Π b:B η(g(b)) = ap(g, ɛ(n)) where ap(f ) : Π a,a ′ :Aa = a ′ → f (a) = f (a ′ ) We can derive δ from δ ′ and vice versa, because equality is a groupoid. We define equivalence A ≈ B as given by isomorphism and δ. Thorsten Altenkirch (Nottingham) Edinburgh 13 October 30, 2013 15 / 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: The univalence axiom Univalence Rej
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 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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