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feat(category_theory): yoneda functor on abelian category preserves f…
…inite colimits iff object is injective (#16893)
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/- | ||
Copyright (c) 2022 Jakob von Raumer. All rights reserved. | ||
Released under Apache 2.0 license as described in the file LICENSE. | ||
Authors: Jakob von Raumer | ||
-/ | ||
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import category_theory.abelian.exact | ||
import category_theory.preadditive.injective | ||
import category_theory.preadditive.yoneda | ||
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/-! | ||
# Injective objects in abelian categories | ||
* Objects in an abelian categories are injective if and only if the preadditive Yoneda functor | ||
on them preserves finite colimits. | ||
-/ | ||
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noncomputable theory | ||
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open category_theory | ||
open category_theory.limits | ||
open category_theory.injective | ||
open opposite | ||
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universes v u | ||
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namespace category_theory | ||
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variables {C : Type u} [category.{v} C] [abelian C] | ||
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/-- The preadditive Yoneda functor on `J` preserves colimits if `J` is injective. -/ | ||
def preserves_finite_colimits_preadditive_yoneda_obj_of_injective (J : C) | ||
[hP : injective J] : preserves_finite_colimits (preadditive_yoneda_obj J) := | ||
begin | ||
letI := (injective_iff_preserves_epimorphisms_preadditive_yoneda_obj' J).mp hP, | ||
apply functor.preserves_finite_colimits_of_preserves_epis_and_kernels, | ||
end | ||
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/-- An object is injective if its preadditive Yoneda functor preserves finite colimits. -/ | ||
lemma injective_of_preserves_finite_colimits_preadditive_yoneda_obj (J : C) | ||
[hP : preserves_finite_colimits (preadditive_yoneda_obj J)] : injective J := | ||
begin | ||
rw injective_iff_preserves_epimorphisms_preadditive_yoneda_obj', | ||
apply_instance | ||
end | ||
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end category_theory |
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