85 lines
2 KiB
Text
85 lines
2 KiB
Text
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/-
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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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Module: types.fiber
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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 types.sigma types.eq
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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
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namespace fiber
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variables {A B : Type} (f : A → B) (b : B)
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definition sigma_char : 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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--set_option pp.notation false
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definition equiv_fiber_eq (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 sigma_char},
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apply equiv.trans,
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{apply equiv.symm, apply equiv_sigma_eq},
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apply sigma_equiv_sigma_id,
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intro p,
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apply equiv_of_equiv_of_eq,
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{apply (ap (λx, x = _)), apply transport_paths_Fl}
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-- apply equiv_of_eq,
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-- fapply (apD011 @sigma),
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-- {apply idp},
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-- esimp
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end
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definition fiber_eq : (p : u.1 = v.1) (q : p ▹ u.2 = v.2) : u = v :=
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by cases u; cases v; apply (dpair_eq_dpair p q)
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end fiber
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namespace is_equiv
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open equiv
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context
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parameters {A B : Type} (f : A → B) [H : is_equiv f]
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include H
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end
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variables {A B : Type} (f : A → B)
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theorem is_hprop_is_equiv [instance] : is_hprop (is_equiv f) :=
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sorry
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end is_equiv
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namespace equiv
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open is_equiv
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variables {A B : Type}
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protected definition eq_mk' {f f' : A → B} [H : is_equiv f] [H' : is_equiv f'] (p : f = f')
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: equiv.mk f H = equiv.mk f' H' :=
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apD011 equiv.mk p !is_hprop.elim
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protected definition eq_mk {f f' : A ≃ B} (p : to_fun f = to_fun f') : f = f' :=
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by (cases f; cases f'; apply (equiv.eq_mk' p))
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end equiv
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