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Mathlib.CategoryTheory.Abelian.Subobject

Equivalence between subobjects and quotients in an abelian category #

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

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