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Mathlib.CategoryTheory.Abelian.Subobject

Equivalence between subobjects and quotients in an abelian category #

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def CategoryTheory.Abelian.subobjectIsoSubobjectOp {C : Type u} [CategoryTheory.Category.{v, u} C] [CategoryTheory.Abelian C] (X : C) :
CategoryTheory.Subobject X ≃o (CategoryTheory.Subobject (Opposite.op X))ᵒᵈ

In an abelian category, the subobjects and quotient objects of an object X are order-isomorphic via taking kernels and cokernels. Implemented here using subobjects in the opposite category, since mathlib does not have a notion of quotient objects at the time of writing.

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Instances For
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    @[simp]
    theorem CategoryTheory.Abelian.subobjectIsoSubobjectOp_symm_apply {C : Type u} [CategoryTheory.Category.{v, u} C] [CategoryTheory.Abelian C] (X : C) (a : (CategoryTheory.Subobject (Opposite.op X))ᵒᵈ) :
    (RelIso.symm (CategoryTheory.Abelian.subobjectIsoSubobjectOp X)) a = CategoryTheory.Subobject.lift (fun (x : Cᵒᵖ) (f : x ⟶ Opposite.op X) (x_1 : CategoryTheory.Mono f) => CategoryTheory.Subobject.mk (CategoryTheory.Limits.kernel.ι f.unop)) ⋯ a
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    @[simp]
    theorem CategoryTheory.Abelian.subobjectIsoSubobjectOp_apply {C : Type u} [CategoryTheory.Category.{v, u} C] [CategoryTheory.Abelian C] (X : C) (a : CategoryTheory.Subobject X) :
    (CategoryTheory.Abelian.subobjectIsoSubobjectOp X) a = CategoryTheory.Subobject.lift (fun (x : C) (f : x ⟶ X) (x_1 : CategoryTheory.Mono f) => CategoryTheory.Subobject.mk (CategoryTheory.Limits.cokernel.π f).op) ⋯ a
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    instance CategoryTheory.Abelian.wellPowered_opposite {C : Type u} [CategoryTheory.Category.{v, u} C] [CategoryTheory.Abelian C] [CategoryTheory.WellPowered C] :
    CategoryTheory.WellPowered Cᵒᵖ

    A well-powered abelian category is also well-copowered.