43 lines
1.5 KiB
Text
43 lines
1.5 KiB
Text
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/-
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Copyright (c) 2016 Egbert Rijke. All rights reserved.
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Released under Apache 2.0 license as described in the file LICENSE.
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Authors: Egbert Rijke
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Exact couple, derived couples, and so on
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-/
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import algebra.group_theory hit.set_quotient types.sigma types.list types.sum
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open eq algebra is_trunc set_quotient relation sigma sigma.ops prod prod.ops sum list trunc function group
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equiv
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definition kernel.{l1} {A B : CommGroup.{l1}} (f : A →g B) : CommGroup.{l1} :=
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begin
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fapply CommGroup.mk,
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{ exact fiber f 1},
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fapply comm_group.mk,
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{ intro x, induction x with a p, intro y, induction y with b q, fapply fiber.mk, exact a*b, rewrite respect_mul, rewrite p, rewrite q, apply mul_one},
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{ exact sorry },
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{ intros x y z, induction x with a p, induction y with b q, induction z with c r, esimp, exact sorry }, repeat exact sorry
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end
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structure is_exact {A B C : CommGroup} (f : A →g B) (g : B →g C) :=
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( im_in_ker : Π(a:A), g (f a) = 1)
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( ker_in_im : Π(b:B), (g b = 1) → Σ(a:A), f a = b)
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definition isBoundary {B : CommGroup} (d : B →g B) := Π(b:B), d b = 1
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-- definition homology {B : CommGroup} (d : B →g B) (H : isBoundary d) :=
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-- quotient_group (kernel d) (image d)
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structure exact_couple (A B : CommGroup) : Type :=
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( i : A →g A) (j : A →g B) (k : B →g A)
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( exact_ij : is_exact i j)
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( exact_jk : is_exact j k)
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( exact_ki : is_exact k i)
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definition boundary {A B : CommGroup} (CC : exact_couple A B) : B →g B := (exact_couple.j CC) ∘g (exact_couple.k CC)
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