138 lines
4.4 KiB
Text
138 lines
4.4 KiB
Text
/-
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Copyright (c) 2015 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 fibers
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-/
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import .sigma .eq .pi .pointed
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structure fiber {A B : Type} (f : A → B) (b : B) :=
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(point : A)
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(point_eq : f point = b)
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open equiv sigma sigma.ops eq pi
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namespace fiber
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variables {A B : Type} {f : A → B} {b : B}
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protected definition sigma_char [constructor]
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(f : A → B) (b : B) : fiber f b ≃ (Σ(a : A), f a = b) :=
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begin
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fapply equiv.MK,
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{intro x, exact ⟨point x, point_eq x⟩},
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{intro x, exact (fiber.mk x.1 x.2)},
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{intro x, cases x, apply idp},
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{intro x, cases x, apply idp},
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end
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definition fiber_eq_equiv (x y : fiber f b)
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: (x = y) ≃ (Σ(p : point x = point y), point_eq x = ap f p ⬝ point_eq y) :=
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begin
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apply equiv.trans,
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apply eq_equiv_fn_eq_of_equiv, apply fiber.sigma_char,
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apply equiv.trans,
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apply sigma_eq_equiv,
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apply sigma_equiv_sigma_id,
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intro p,
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apply pathover_eq_equiv_Fl,
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end
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definition fiber_eq {x y : fiber f b} (p : point x = point y)
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(q : point_eq x = ap f p ⬝ point_eq y) : x = y :=
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to_inv !fiber_eq_equiv ⟨p, q⟩
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open is_trunc
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definition fiber_pr1 (B : A → Type) (a : A) : fiber (pr1 : (Σa, B a) → A) a ≃ B a :=
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calc
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fiber pr1 a ≃ Σu, u.1 = a : fiber.sigma_char
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... ≃ Σa' (b : B a'), a' = a : sigma_assoc_equiv
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... ≃ Σa' (p : a' = a), B a' : sigma_equiv_sigma_id (λa', !comm_equiv_nondep)
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... ≃ Σu, B u.1 : sigma_assoc_equiv
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... ≃ B a : !sigma_equiv_of_is_contr_left
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definition sigma_fiber_equiv (f : A → B) : (Σb, fiber f b) ≃ A :=
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calc
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(Σb, fiber f b) ≃ Σb a, f a = b : sigma_equiv_sigma_id (λb, !fiber.sigma_char)
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... ≃ Σa b, f a = b : sigma_comm_equiv
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... ≃ A : sigma_equiv_of_is_contr_right
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definition is_pointed_fiber [instance] [constructor] (f : A → B) (a : A)
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: pointed (fiber f (f a)) :=
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pointed.mk (fiber.mk a idp)
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definition pointed_fiber [constructor] (f : A → B) (a : A) : Type* :=
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Pointed.mk (fiber.mk a (idpath (f a)))
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end fiber
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open unit is_trunc
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namespace fiber
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definition fiber_star_equiv (A : Type) : fiber (λx : A, star) star ≃ A :=
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begin
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fapply equiv.MK,
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{ intro f, cases f with a H, exact a },
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{ intro a, apply fiber.mk a, reflexivity },
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{ intro a, reflexivity },
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{ intro f, cases f with a H, change fiber.mk a (refl star) = fiber.mk a H,
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rewrite [is_hset.elim H (refl star)] }
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end
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definition fiber_const_equiv (A : Type) (a₀ : A) (a : A)
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: fiber (λz : unit, a₀) a ≃ a₀ = a :=
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calc
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fiber (λz : unit, a₀) a
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≃ Σz : unit, a₀ = a : fiber.sigma_char
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... ≃ a₀ = a : sigma_unit_left
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end fiber
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open function is_equiv
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namespace fiber
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/- Theorem 4.7.6 -/
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variables {A : Type} {P Q : A → Type}
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variable (f : Πa, P a → Q a)
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/- Note that the map on total spaces/sigmas is just sigma_functor id -/
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definition fiber_total_equiv {a : A} (q : Q a)
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: fiber (sigma_functor id f) ⟨a , q⟩ ≃ fiber (f a) q :=
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calc
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fiber (sigma_functor id f) ⟨a , q⟩
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≃ Σ(w : Σx, P x), ⟨w.1 , f w.1 w.2 ⟩ = ⟨a , q⟩
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: fiber.sigma_char
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... ≃ Σ(x : A), Σ(p : P x), ⟨x , f x p⟩ = ⟨a , q⟩
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: sigma_assoc_equiv
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... ≃ Σ(x : A), Σ(p : P x), Σ(H : x = a), f x p =[H] q
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:
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begin
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apply sigma_equiv_sigma_id, intro x,
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apply sigma_equiv_sigma_id, intro p,
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apply sigma_eq_equiv
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end
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... ≃ Σ(x : A), Σ(H : x = a), Σ(p : P x), f x p =[H] q
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:
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begin
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apply sigma_equiv_sigma_id, intro x,
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apply sigma_comm_equiv
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end
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... ≃ Σ(w : Σx, x = a), Σ(p : P w.1), f w.1 p =[w.2] q
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: sigma_assoc_equiv
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... ≃ Σ(p : P (center (Σx, x=a)).1), f (center (Σx, x=a)).1 p =[(center (Σx, x=a)).2] q
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: sigma_equiv_of_is_contr_left
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... ≃ Σ(p : P a), f a p =[idpath a] q
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: equiv_of_eq idp
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... ≃ Σ(p : P a), f a p = q
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:
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begin
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apply sigma_equiv_sigma_id, intro p,
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apply pathover_idp
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end
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... ≃ fiber (f a) q
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: fiber.sigma_char
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end fiber
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