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feat(category_theory/abelian): pseudoelements and a four lemma (#3803)
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/- | ||
Copyright (c) 2020 Markus Himmel. All rights reserved. | ||
Released under Apache 2.0 license as described in the file LICENSE. | ||
Authors: Markus Himmel | ||
-/ | ||
import category_theory.abelian.pseudoelements | ||
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/-! | ||
# The four lemma | ||
Consider the following commutative diagram with exact rows in an abelian category: | ||
A ---f--> B ---g--> C ---h--> D | ||
| | | | | ||
α β γ δ | ||
| | | | | ||
v v v v | ||
A' --f'-> B' --g'-> C' --h'-> D' | ||
We prove the "mono" version of the four lemma: if α is an epimorphism and β and δ are monomorphisms, | ||
then γ is a monomorphism. | ||
## Future work | ||
The "epi" four lemma and the five lemma, which is then an easy corollary. | ||
## Tags | ||
four lemma, diagram lemma, diagram chase | ||
-/ | ||
open category_theory | ||
open category_theory.abelian.pseudoelement | ||
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universes v u | ||
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variables {V : Type u} [category.{v} V] [abelian V] | ||
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local attribute [instance] preadditive.has_equalizers_of_has_kernels | ||
local attribute [instance] object_to_sort hom_to_fun | ||
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namespace category_theory.abelian | ||
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variables {A B C D A' B' C' D' : V} | ||
variables {f : A ⟶ B} {g : B ⟶ C} {h : C ⟶ D} | ||
variables {f' : A' ⟶ B'} {g' : B' ⟶ C'} {h' : C' ⟶ D'} | ||
variables {α : A ⟶ A'} {β : B ⟶ B'} {γ : C ⟶ C'} {δ : D ⟶ D'} | ||
variables [exact f g] [exact g h] [exact f' g'] | ||
variables (comm₁ : α ≫ f' = f ≫ β) (comm₂ : β ≫ g' = g ≫ γ) (comm₃ : γ ≫ h' = h ≫ δ) | ||
include comm₁ comm₂ comm₃ | ||
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/-- The four lemma, mono version. For names of objects and morphisms, consider the following | ||
diagram: | ||
``` | ||
A ---f--> B ---g--> C ---h--> D | ||
| | | | | ||
α β γ δ | ||
| | | | | ||
v v v v | ||
A' --f'-> B' --g'-> C' --h'-> D' | ||
``` | ||
-/ | ||
lemma mono_of_epi_of_mono_of_mono (hα : epi α) (hβ : mono β) (hδ : mono δ) : mono γ := | ||
mono_of_zero_of_map_zero _ $ λ c hc, | ||
have h c = 0, from | ||
suffices δ (h c) = 0, from zero_of_map_zero _ (pseudo_injective_of_mono _) _ this, | ||
calc δ (h c) = h' (γ c) : by rw [←comp_apply, ←comm₃, comp_apply] | ||
... = h' 0 : by rw hc | ||
... = 0 : apply_zero _, | ||
exists.elim (pseudo_exact_of_exact.2 _ this) $ λ b hb, | ||
have g' (β b) = 0, from | ||
calc g' (β b) = γ (g b) : by rw [←comp_apply, comm₂, comp_apply] | ||
... = γ c : by rw hb | ||
... = 0 : hc, | ||
exists.elim (pseudo_exact_of_exact.2 _ this) $ λ a' ha', | ||
exists.elim (pseudo_surjective_of_epi α a') $ λ a ha, | ||
have f a = b, from | ||
suffices β (f a) = β b, from pseudo_injective_of_mono _ this, | ||
calc β (f a) = f' (α a) : by rw [←comp_apply, ←comm₁, comp_apply] | ||
... = f' a' : by rw ha | ||
... = β b : ha', | ||
calc c = g b : hb.symm | ||
... = g (f a) : by rw this | ||
... = 0 : pseudo_exact_of_exact.1 _ | ||
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end category_theory.abelian |
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