Equivalence between subobjects and quotients in an abelian category

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.

Equations

• CategoryTheory.Abelian.subobjectIsoSubobjectOp X = OrderIso.ofHomInv (CategoryTheory.Limits.cokernelOrderHom X) (CategoryTheory.Limits.kernelOrderHom X) ⋯ ⋯

@[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

@[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

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.

Equations

• ⋯ = ⋯