factor free group on set via free group on pointed set
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Ulrik Buchholtz
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2 changed files with 59 additions and 18 deletions
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@ -1,7 +1,7 @@
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/-
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Copyright (c) 2015 Floris van Doorn. All rights reserved.
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Copyright (c) 2015-2018 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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Authors: Floris van Doorn, Egbert Rijke
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Authors: Floris van Doorn, Egbert Rijke, Ulrik Buchholtz
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Constructions with groups
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-/
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@ -9,7 +9,7 @@ Constructions with groups
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import algebra.group_theory hit.set_quotient types.list types.sum ..move_to_lib
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open eq algebra is_trunc set_quotient relation sigma sigma.ops prod sum list trunc function equiv
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prod.ops decidable is_equiv
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prod.ops decidable is_equiv pointed
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universe variable u
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@ -17,14 +17,16 @@ namespace group
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variables {G G' : Group} {g g' h h' k : G} {A B : AbGroup}
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/- Free Group of a set -/
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variables (X : Type) [is_set X] {l l' : list (X ⊎ X)}
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/- Free Group of a pointed set -/
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variables (X : Type*) [is_set X] {l l' : list (X ⊎ X)}
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namespace free_group
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inductive free_group_rel : list (X ⊎ X) → list (X ⊎ X) → Type :=
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| rrefl : Πl, free_group_rel l l
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| cancel1 : Πx, free_group_rel [inl x, inr x] []
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| cancel2 : Πx, free_group_rel [inr x, inl x] []
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| cancelpt1 : free_group_rel [inl pt] []
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| cancelpt2 : free_group_rel [inr pt] []
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| resp_append : Π{l₁ l₂ l₃ l₄}, free_group_rel l₁ l₂ → free_group_rel l₃ l₄ →
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free_group_rel (l₁ ++ l₃) (l₂ ++ l₄)
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| rtrans : Π{l₁ l₂ l₃}, free_group_rel l₁ l₂ → free_group_rel l₂ l₃ →
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@ -48,6 +50,8 @@ namespace group
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{ reflexivity},
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{ repeat esimp [map], exact cancel2 x},
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{ repeat esimp [map], exact cancel1 x},
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{ exact cancelpt2 X },
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{ exact cancelpt1 X },
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{ rewrite [+map_append], exact resp_append IH₁ IH₂},
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{ exact rtrans IH₁ IH₂}
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end
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@ -58,6 +62,8 @@ namespace group
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{ reflexivity},
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{ repeat esimp [map], exact cancel2 x},
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{ repeat esimp [map], exact cancel1 x},
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{ exact cancelpt1 X },
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{ exact cancelpt2 X },
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{ rewrite [+reverse_append], exact resp_append IH₂ IH₁},
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{ exact rtrans IH₁ IH₂}
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end
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@ -127,15 +133,17 @@ namespace group
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class_of [inl x]
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definition fgh_helper [unfold 6] (f : X → G) (g : G) (x : X ⊎ X) : G :=
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g * sum.rec (λx, f x) (λx, (f x)⁻¹) x
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g * sum.rec (λz, f z) (λz, (f z)⁻¹) x
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theorem fgh_helper_respect_rel (f : X → G) (r : free_group_rel X l l')
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theorem fgh_helper_respect_rel (f : X →* G) (r : free_group_rel X l l')
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: Π(g : G), foldl (fgh_helper f) g l = foldl (fgh_helper f) g l' :=
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begin
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induction r with l x x l₁ l₂ l₃ l₄ r₁ r₂ IH₁ IH₂ l₁ l₂ l₃ r₁ r₂ IH₁ IH₂: intro g,
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{ reflexivity},
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{ unfold [foldl], apply mul_inv_cancel_right},
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{ unfold [foldl], apply inv_mul_cancel_right},
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{ unfold [foldl], rewrite (respect_pt f), apply mul_one },
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{ unfold [foldl], rewrite [respect_pt f, one_inv], apply mul_one },
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{ rewrite [+foldl_append, IH₁, IH₂]},
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{ exact !IH₁ ⬝ !IH₂}
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end
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@ -149,7 +157,7 @@ namespace group
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rewrite [-mul.assoc, ↑fgh_helper, one_mul]}
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end
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definition free_group_hom [constructor] (f : X → G) : free_group X →g G :=
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definition free_group_hom [constructor] (f : X →* G) : free_group X →g G :=
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begin
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fapply homomorphism.mk,
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{ intro g, refine set_quotient.elim _ _ g,
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@ -173,16 +181,23 @@ namespace group
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(respect_inv ψ (class_of [inl x]))⁻¹ }
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end
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definition fn_of_free_group_hom [unfold_full] (φ : free_group X →g G) : X → G :=
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φ ∘ free_group_inclusion
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definition fn_of_free_group_hom [unfold_full] (φ : free_group X →g G) : X →* G :=
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ppi.mk (φ ∘ free_group_inclusion)
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begin
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refine (_ ⬝ respect_one φ),
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apply ap φ, apply eq_of_rel, apply tr,
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exact (free_group_rel.cancelpt1 X)
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end
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variables (X G)
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definition free_group_hom_equiv_fn : (free_group X →g G) ≃ (X → G) :=
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definition free_group_hom_equiv_fn : (free_group X →g G) ≃ (X →* G) :=
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begin
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fapply equiv.MK,
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{ exact fn_of_free_group_hom},
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{ exact free_group_hom},
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{ intro f, apply eq_of_homotopy, intro x, esimp, unfold [foldl], apply one_mul},
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{ intro f, apply eq_of_phomotopy, fapply phomotopy.mk,
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{ intro x, esimp, unfold [foldl], apply one_mul },
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{ apply is_prop.elim } },
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{ intro φ, apply homomorphism_eq, apply free_group_hom_eq, intro x, apply one_mul }
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end
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@ -629,16 +644,20 @@ namespace group
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variable (X)
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definition free_group_of_dfree_group [constructor] : dfree_group X →g free_group X :=
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dfree_group_hom free_group_inclusion
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open option
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definition dfree_group_of_free_group [constructor] : free_group X →g dfree_group X :=
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free_group_hom dfree_group_inclusion
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definition free_group_of_dfree_group [constructor] : dfree_group X →g free_group X₊ :=
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dfree_group_hom (free_group_inclusion ∘ some)
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definition dfree_group_isomorphism : dfree_group X ≃g free_group X :=
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definition dfree_group_of_free_group [constructor] : free_group X₊ →g dfree_group X :=
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free_group_hom (ppi.mk (option.rec 1 dfree_group_inclusion) idp)
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definition dfree_group_isomorphism : dfree_group X ≃g free_group X₊ :=
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begin
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apply isomorphism.MK (free_group_of_dfree_group X) (dfree_group_of_free_group X),
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{ apply free_group_hom_eq, intro x, reflexivity },
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{ apply free_group_hom_eq, intro x, induction x with x,
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{ symmetry, apply eq_of_rel, apply tr, exact free_group.free_group_rel.cancelpt1 X₊ },
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{ reflexivity } },
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{ apply dfree_group_hom_eq, intro x, reflexivity }
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end
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@ -69,6 +69,28 @@ end nat
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namespace pointed
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open option sum
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definition option_equiv_sum (A : Type) : option A ≃ A ⊎ unit :=
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begin
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fapply equiv.MK,
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{ intro z, induction z with a,
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{ exact inr star },
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{ exact inl a } },
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{ intro z, induction z with a b,
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{ exact some a },
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{ exact none } },
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{ intro z, induction z with a b,
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{ reflexivity },
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{ induction b, reflexivity } },
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{ intro z, induction z with a, all_goals reflexivity }
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end
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theorem is_trunc_add_point [instance] (n : ℕ₋₂) (A : Type) [HA : is_trunc (n.+2) A]
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: is_trunc (n.+2) A₊ :=
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begin
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apply is_trunc_equiv_closed_rev _ (option_equiv_sum A),
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apply is_trunc_sum
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end
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end pointed open pointed
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