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Since each P(x) is contractible there is an equivalence P(x) ≃ unit for each x ∶ A. By the univalence axiom we get a path P(x) ↝ unit. Since we have a path P ↝ U, there is a path between the types ( ∏(x ∶ A), P(x)) ↝ ( ∏(x ∶ A), U(x)) so we only have to verify that ∏(x ∶ A), U(x), which is equal to A → unit, is contractible. 6. We finish this section with two new applications of the univalence axiom. The first is an observation of Bas Spitters, which he shared with the author at the Fourth Workshop on Formal Topology in Ljubljana in 2012.

Using the function extensionality principle, it suffices to show that ∏(x ∶ unit), ϕ( f (tt))(x) ↝ f (x). To show that such a section exists, we use the induction principle for unit once more: note that ϕ( f (tt))(tt) = f (tt). 6. For every space A there is an equivalence A ≃ (unit → A). 7 (Correspondence theorem for empty). Suppose P is a dependent type over the empty type empty. Then we have an equivalence unit ≃ ∏(x ∶ empty), P(x). 45 P ROOF. Note that it suffices to show that ∏(x ∶ empty), P(x) is contractible.

Proj1 ⟨ f (x),⟨x,id f (x) ⟩⟩ = λ x. f (x) = f , which finishes the proof. The second application of the univalence axiom asserts that the notions of function and graph coincide. We have chosen to state the theorem in its non-dependent form, but the proof of the dependent version of the assertion goes along the same lines. 10. e. there is an equivalence ( ∑(R ∶ A → B → Type) ∏(a ∶ A), isContr( ∑(b ∶ B), R(a,b))) ≃ A → B. P ROOF. For any function f ∶ A → B there is the relation R f ∶ A → B → Type given by R f (a,b) ∶= f (a) ↝ b.

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