Coskeletal simplicial objects #

The identity natural transformation exhibits a simplicial object X as a right extension of its restriction along (Truncated.inclusion n).op recorded by rightExtensionInclusion X n.

The simplicial object X is n-coskeletal if rightExtensionInclusion X n is a right Kan extension.

When the ambient category admits right Kan extensions along (Truncated.inclusion n).op, then when X is n-coskeletal, the unit of coskAdj n defines an isomorphism: isoCoskOfIsCoskeletal : X ≅ (cosk n).obj X.

TODO: Prove that X is n-coskeletal whenever a certain canonical cone is a limit cone.

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def CategoryTheory.SimplicialObject.Truncated.rightExtensionInclusion {C : Type u} [CategoryTheory.Category.{v, u} C] (X : CategoryTheory.SimplicialObject C) (n : ℕ) :

(SimplexCategory.Truncated.inclusion n).op.RightExtension ((SimplexCategory.Truncated.inclusion n).op.comp X)

The identity natural transformation exhibits a simplicial set as a right extension of its restriction along (Truncated.inclusion n).op.

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@[simp]

theorem CategoryTheory.SimplicialObject.Truncated.rightExtensionInclusion_hom_app {C : Type u} [CategoryTheory.Category.{v, u} C] (X : CategoryTheory.SimplicialObject C) (n : ℕ) (X✝ : (SimplexCategory.Truncated n)ᵒᵖ) :

(CategoryTheory.SimplicialObject.Truncated.rightExtensionInclusion X n).hom.app X✝ = CategoryTheory.CategoryStruct.id (X.obj (Opposite.op ((SimplexCategory.Truncated.inclusion n).obj (Opposite.unop X✝))))

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@[simp]

theorem CategoryTheory.SimplicialObject.Truncated.rightExtensionInclusion_left {C : Type u} [CategoryTheory.Category.{v, u} C] (X : CategoryTheory.SimplicialObject C) (n : ℕ) :

(CategoryTheory.SimplicialObject.Truncated.rightExtensionInclusion X n).left = X

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theorem CategoryTheory.SimplicialObject.Truncated.rightExtensionInclusion_right_as {C : Type u} [CategoryTheory.Category.{v, u} C] (X : CategoryTheory.SimplicialObject C) (n : ℕ) :

(CategoryTheory.SimplicialObject.Truncated.rightExtensionInclusion X n).right.as = PUnit.unit

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class CategoryTheory.SimplicialObject.IsCoskeletal {C : Type u} [CategoryTheory.Category.{v, u} C] (X : CategoryTheory.SimplicialObject C) (n : ℕ) :

Prop

A simplicial object X is n-coskeletal when it is the right Kan extension of its restriction along (Truncated.inclusion n).op via the identity natural transformation.

isRightKanExtension : CategoryTheory.Functor.IsRightKanExtension X (CategoryTheory.CategoryStruct.id ((SimplexCategory.Truncated.inclusion n).op.comp X))

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theorem CategoryTheory.SimplicialObject.isCoskeletal_iff {C : Type u} [CategoryTheory.Category.{v, u} C] (X : CategoryTheory.SimplicialObject C) (n : ℕ) :

X.IsCoskeletal n ↔ CategoryTheory.Functor.IsRightKanExtension X (CategoryTheory.CategoryStruct.id ((SimplexCategory.Truncated.inclusion n).op.comp X))

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noncomputable def CategoryTheory.SimplicialObject.IsCoskeletal.isUniversalOfIsRightKanExtension {C : Type u} [CategoryTheory.Category.{v, u} C] (X : CategoryTheory.SimplicialObject C) (n : ℕ) [X.IsCoskeletal n] :

CategoryTheory.CostructuredArrow.IsUniversal (CategoryTheory.SimplicialObject.Truncated.rightExtensionInclusion X n)

If X is n-coskeletal, then Truncated.rightExtensionInclusion X n is a terminal object in the category RightExtension (Truncated.inclusion n).op (Truncated.inclusion.op ⋙ X).

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