66 lines
2.3 KiB
Text
66 lines
2.3 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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Authors: Floris van Doorn, Egbert Rijke
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Constructions with groups
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-/
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import .quotient_group .free_commutative_group
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open eq algebra is_trunc set_quotient relation sigma prod sum list trunc function equiv sigma.ops
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namespace group
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variables {G G' : Group} (H : subgroup_rel G) (N : normal_subgroup_rel G) {g g' h h' k : G}
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{A B : AbGroup}
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variables (X : Set) {l l' : list (X ⊎ X)}
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section
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parameters {I : Set} (Y : I → AbGroup)
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variables {A' : AbGroup}
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definition dirsum_carrier : AbGroup := free_ab_group (trunctype.mk (Σi, Y i) _)
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local abbreviation ι [constructor] := @free_ab_group_inclusion
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inductive dirsum_rel : dirsum_carrier → Type :=
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| rmk : Πi y₁ y₂, dirsum_rel (ι ⟨i, y₁⟩ * ι ⟨i, y₂⟩ * (ι ⟨i, y₁ * y₂⟩)⁻¹)
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definition dirsum : AbGroup := quotient_ab_group_gen dirsum_carrier (λg, ∥dirsum_rel g∥)
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-- definition dirsum_carrier_incl [constructor] (i : I) : Y i →g dirsum_carrier :=
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definition dirsum_incl [constructor] (i : I) : Y i →g dirsum :=
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homomorphism.mk (λy, class_of (ι ⟨i, y⟩))
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begin intro g h, symmetry, apply gqg_eq_of_rel, apply tr, apply dirsum_rel.rmk end
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definition dirsum_elim_resp_quotient (f : Πi, Y i →g A') (g : dirsum_carrier)
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(r : ∥dirsum_rel g∥) : free_ab_group_elim (λv, f v.1 v.2) g = 1 :=
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begin
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induction r with r, induction r,
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rewrite [to_respect_mul, to_respect_inv], apply mul_inv_eq_of_eq_mul,
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rewrite [one_mul, to_respect_mul, ▸*, ↑foldl, +one_mul, to_respect_mul]
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end
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definition dirsum_elim [constructor] (f : Πi, Y i →g A') : dirsum →g A' :=
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gqg_elim _ (free_ab_group_elim (λv, f v.1 v.2)) (dirsum_elim_resp_quotient f)
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definition dirsum_elim_compute (f : Πi, Y i →g A') (i : I) :
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dirsum_elim f ∘g dirsum_incl i ~ f i :=
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begin
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intro g, apply one_mul
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end
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definition dirsum_elim_unique (f : Πi, Y i →g A') (k : dirsum →g A')
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(H : Πi, k ∘g dirsum_incl i ~ f i) : k ~ dirsum_elim f :=
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begin
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apply gqg_elim_unique,
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apply free_ab_group_elim_unique,
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intro x, induction x with i y, exact H i y
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end
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end
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end group
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