89 lines
2 KiB
Text
89 lines
2 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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Module: algebra.fundamental_group
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Authors: Floris van Doorn
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fundamental group of a pointed space
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-/
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import hit.trunc algebra.group types.pointed
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open core eq trunc algebra is_trunc pointed
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namespace fundamental_group
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section
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parameters {A : Type} (a : A)
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definition carrier [reducible] : Type :=
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trunc 0 (a = a)
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local abbreviation G := carrier
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definition mul (g h : G) : G :=
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begin
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apply trunc.rec_on g, intro p,
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apply trunc.rec_on h, intro q,
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exact tr (p ⬝ q)
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end
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definition inv (g : G) : G :=
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begin
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apply trunc.rec_on g, intro p,
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exact tr p⁻¹
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end
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definition one : G :=
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tr idp
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local notation 1 := one
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local postfix ⁻¹ := inv
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local infix * := mul
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definition mul_assoc (g₁ g₂ g₃ : G) : g₁ * g₂ * g₃ = g₁ * (g₂ * g₃) :=
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begin
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apply trunc.rec_on g₁, intro p₁,
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apply trunc.rec_on g₂, intro p₂,
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apply trunc.rec_on g₃, intro p₃,
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exact ap tr !con.assoc,
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end
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definition one_mul (g : G) : 1 * g = g :=
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begin
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apply trunc.rec_on g, intro p,
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exact ap tr !idp_con,
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end
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definition mul_one (g : G) : g * 1 = g :=
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begin
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apply trunc.rec_on g, intro p,
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exact idp,
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end
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definition mul_left_inv (g : G) : g⁻¹ * g = 1 :=
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begin
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apply trunc.rec_on g, intro p,
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apply ap tr !con.left_inv
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end
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definition group : group G :=
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⦃group,
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mul := mul,
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mul_assoc := mul_assoc,
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one := one,
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one_mul := one_mul,
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mul_one := mul_one,
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inv := inv,
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mul_left_inv := mul_left_inv,
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is_hset_carrier := _⦄
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end
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end fundamental_group
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attribute fundamental_group.group [instance] [constructor] [priority 800]
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definition fundamental_group [constructor] (A : Type) [H : pointed A]: Group :=
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Group.mk (fundamental_group.carrier (point A)) _
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namespace core
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prefix `π₁`:95 := fundamental_group
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end core
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