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feat(Algebra/Category/GroupCat/Abelian): prove AddCommGroupCat is AB5 (…
…#5597) This work was done during the 2023 Copenhagen masterclass on formalisation of condensed mathematics. Numerous participants contributed. Co-authored-by: Moritz Firsching <moritz.firsching@gmail.com> Co-authored-by: Nikolas Kuhn <nikolaskuhn@gmx.de> Co-authored-by: Amelia Livingston <101damnations@github.com> Co-authored-by: Markus Himmel <markus@himmel-villmar.de> Co-authored-by: Moritz Firsching <firsching@google.com> Co-authored-by: adamtopaz <github@adamtopaz.com> Co-authored-by: nick-kuhn <nikolaskuhn@gmx.de>
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/- | ||
Copyright (c) 2023 Moritz Firsching. All rights reserved. | ||
Released under Apache 2.0 license as described in the file LICENSE. | ||
Authors: David Kurniadi Angdinata, Moritz Firsching, Nikolas Kuhn | ||
-/ | ||
import Mathlib.Algebra.Category.GroupCat.EpiMono | ||
import Mathlib.Algebra.Category.GroupCat.Preadditive | ||
import Mathlib.CategoryTheory.Limits.Shapes.Kernels | ||
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/-! | ||
# The concrete (co)kernels in the category of abelian groups are categorical (co)kernels. | ||
-/ | ||
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namespace AddCommGroupCat | ||
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open AddMonoidHom CategoryTheory Limits QuotientAddGroup | ||
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universe u | ||
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variable {G H : AddCommGroupCat.{u}} (f : G ⟶ H) | ||
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/-- The kernel cone induced by the concrete kernel. -/ | ||
def kernelCone : KernelFork f := | ||
KernelFork.ofι (Z := of f.ker) f.ker.subtype <| ext fun x => x.casesOn fun _ hx => hx | ||
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/-- The kernel of a group homomorphism is a kernel in the categorical sense. -/ | ||
def kernelIsLimit : IsLimit <| kernelCone f := | ||
Fork.IsLimit.mk _ | ||
(fun s => (by exact Fork.ι s : _ →+ G).codRestrict _ <| fun c => f.mem_ker.mpr <| | ||
by exact FunLike.congr_fun s.condition c) | ||
(fun _ => by rfl) | ||
(fun _ _ h => ext fun x => Subtype.ext_iff_val.mpr <| by exact FunLike.congr_fun h x) | ||
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/-- The cokernel cocone induced by the projection onto the quotient. -/ | ||
def cokernelCocone : CokernelCofork f := | ||
CokernelCofork.ofπ (Z := of <| H ⧸ f.range) (mk' f.range) <| ext fun x => | ||
(eq_zero_iff _).mpr ⟨x, rfl⟩ | ||
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/-- The projection onto the quotient is a cokernel in the categorical sense. -/ | ||
def cokernelIsColimit : IsColimit <| cokernelCocone f := | ||
Cofork.IsColimit.mk _ | ||
(fun s => lift _ _ <| (range_le_ker_iff _ _).mpr <| CokernelCofork.condition s) | ||
(fun _ => rfl) | ||
(fun _ _ h => have : Epi (cokernelCocone f).π := (epi_iff_surjective _).mpr <| mk'_surjective _ | ||
(cancel_epi _).mp <| by simpa only [parallelPair_obj_one] using h) | ||
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end AddCommGroupCat |
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