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