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feat(category_theory/abelian/*): functors that preserve finite limits…
… and colimits preserve exactness (#14581) If $F$ is a functor between two abelian categories which preserves limits and colimits, then it preserves exactness.
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/- | ||
Copyright (c) 2022 Jujian Zhang. All rights reserved. | ||
Released under Apache 2.0 license as described in the file LICENSE. | ||
Authors: Jujian Zhang | ||
-/ | ||
import category_theory.limits.shapes.images | ||
import category_theory.limits.constructions.epi_mono | ||
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/-! | ||
# Preserving images | ||
In this file, we show that if a functor preserves span and cospan, then it preserves images. | ||
-/ | ||
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noncomputable theory | ||
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namespace category_theory | ||
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namespace preserves_image | ||
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open category_theory | ||
open category_theory.limits | ||
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universes u₁ u₂ v₁ v₂ | ||
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variables {A : Type u₁} {B : Type u₂} [category.{v₁} A] [category.{v₂} B] | ||
variables [has_equalizers A] [has_images A] | ||
variables [strong_epi_category B] [has_images B] | ||
variables (L : A ⥤ B) | ||
variables [Π {X Y Z : A} (f : X ⟶ Z) (g : Y ⟶ Z), preserves_limit (cospan f g) L] | ||
variables [Π {X Y Z : A} (f : X ⟶ Y) (g : X ⟶ Z), preserves_colimit (span f g) L] | ||
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/-- | ||
If a functor preserves span and cospan, then it preserves images. | ||
-/ | ||
@[simps] def iso {X Y : A} (f : X ⟶ Y) : image (L.map f) ≅ L.obj (image f) := | ||
let aux1 : strong_epi_mono_factorisation (L.map f) := | ||
{ I := L.obj (limits.image f), | ||
m := L.map $ limits.image.ι _, | ||
m_mono := infer_instance, | ||
e := L.map $ factor_thru_image _, | ||
e_strong_epi := @@strong_epi_of_epi _ _ _ $ | ||
@@category_theory.preserves_epi _ _ L _ _ _, | ||
fac' := by rw [←L.map_comp, limits.image.fac] } in | ||
is_image.iso_ext (image.is_image (L.map f)) aux1.to_mono_is_image | ||
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@[reassoc] lemma factor_thru_image_comp_hom {X Y : A} (f : X ⟶ Y) : | ||
factor_thru_image (L.map f) ≫ (iso L f).hom = | ||
L.map (factor_thru_image f) := | ||
by simp | ||
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@[reassoc] lemma hom_comp_map_image_ι {X Y : A} (f : X ⟶ Y) : | ||
(iso L f).hom ≫ L.map (image.ι f) = image.ι (L.map f) := | ||
by simp | ||
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@[reassoc] lemma inv_comp_image_ι_map {X Y : A} (f : X ⟶ Y) : | ||
(iso L f).inv ≫ image.ι (L.map f) = L.map (image.ι f) := | ||
by simp | ||
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end preserves_image | ||
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end category_theory |
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