7e52c49dce
Prove that 'is_left_adjoint F' is a mere proposition, although this proof is commented out because it takes ~10 seconds
86 lines
2.8 KiB
Text
86 lines
2.8 KiB
Text
/-
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Copyright (c) 2014 Floris van Doorn. All rights reserved.
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Released under Apache 2.0 license as described in the file LICENSE.
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Author: Floris van Doorn
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Ported from Coq HoTT
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Theorems about arrow types (function spaces)
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-/
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import types.pi
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open eq equiv is_equiv funext pi equiv.ops
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namespace pi
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variables {A A' : Type} {B B' : Type} {C : A → B → Type}
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{a a' a'' : A} {b b' b'' : B} {f g : A → B}
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-- all lemmas here are special cases of the ones for pi-types
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/- Functorial action -/
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variables (f0 : A' → A) (f1 : B → B')
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definition arrow_functor : (A → B) → (A' → B') := pi_functor f0 (λa, f1)
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/- Equivalences -/
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definition is_equiv_arrow_functor
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[H0 : is_equiv f0] [H1 : is_equiv f1] : is_equiv (arrow_functor f0 f1) :=
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is_equiv_pi_functor f0 (λa, f1)
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definition arrow_equiv_arrow_rev (f0 : A' ≃ A) (f1 : B ≃ B') : (A → B) ≃ (A' → B') :=
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equiv.mk _ (is_equiv_arrow_functor f0 f1)
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definition arrow_equiv_arrow (f0 : A ≃ A') (f1 : B ≃ B') : (A → B) ≃ (A' → B') :=
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arrow_equiv_arrow_rev (equiv.symm f0) f1
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definition arrow_equiv_arrow_right (f1 : B ≃ B') : (A → B) ≃ (A → B') :=
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arrow_equiv_arrow_rev equiv.refl f1
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definition arrow_equiv_arrow_left_rev (f0 : A' ≃ A) : (A → B) ≃ (A' → B) :=
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arrow_equiv_arrow_rev f0 equiv.refl
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definition arrow_equiv_arrow_left (f0 : A ≃ A') : (A → B) ≃ (A' → B) :=
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arrow_equiv_arrow f0 equiv.refl
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definition arrow_equiv_arrow_right' (f1 : A → (B ≃ B')) : (A → B) ≃ (A → B') :=
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pi_equiv_pi_id f1
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/- Transport -/
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definition arrow_transport {B C : A → Type} (p : a = a') (f : B a → C a)
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: (transport (λa, B a → C a) p f) ~ (λb, p ▸ f (p⁻¹ ▸ b)) :=
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eq.rec_on p (λx, idp)
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/- Pathovers -/
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definition arrow_pathover {B C : A → Type} {f : B a → C a} {g : B a' → C a'} {p : a = a'}
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(r : Π(b : B a) (b' : B a') (q : b =[p] b'), f b =[p] g b') : f =[p] g :=
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begin
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cases p, apply pathover_idp_of_eq,
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apply eq_of_homotopy, intro b,
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exact eq_of_pathover_idp (r b b idpo),
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end
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definition arrow_pathover_left {B C : A → Type} {f : B a → C a} {g : B a' → C a'} {p : a = a'}
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(r : Π(b : B a), f b =[p] g (p ▸ b)) : f =[p] g :=
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begin
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cases p, apply pathover_idp_of_eq,
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apply eq_of_homotopy, intro b,
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exact eq_of_pathover_idp (r b),
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end
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definition arrow_pathover_right {B C : A → Type} {f : B a → C a} {g : B a' → C a'} {p : a = a'}
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(r : Π(b' : B a'), f (p⁻¹ ▸ b') =[p] g b') : f =[p] g :=
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begin
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cases p, apply pathover_idp_of_eq,
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apply eq_of_homotopy, intro b,
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exact eq_of_pathover_idp (r b),
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end
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definition arrow_pathover_constant {B : Type} {C : A → Type} {f : B → C a} {g : B → C a'}
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{p : a = a'} (r : Π(b : B), f b =[p] g b) : f =[p] g :=
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pi_pathover_constant r
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end pi
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