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Mathlib.CategoryTheory.Limits.Yoneda

Limit properties relating to the (co)yoneda embedding. #

We calculate the colimit of Y ↦ (X ⟶ Y), which is just PUnit. (This is used in characterising cofinal functors.)

We also show the (co)yoneda embeddings preserve limits and jointly reflect them.

@[simp]
theorem CategoryTheory.Coyoneda.colimitCocone_ι_app {C : Type u} [CategoryTheory.Category.{v, u} C] (X : Cᵒᵖ) (X_1 : C) (a : (CategoryTheory.coyoneda.obj X).obj X_1) :

The colimit cocone over coyoneda.obj X, with cocone point PUnit.

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    The proposed colimit cocone over coyoneda.obj X is a colimit cocone.

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      The colimit of coyoneda.obj X is isomorphic to PUnit.

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        def CategoryTheory.Limits.coneOfSectionCompYoneda {C : Type u} [CategoryTheory.Category.{v, u} C] {J : Type w} [CategoryTheory.Category.{t, w} J] (F : CategoryTheory.Functor J Cᵒᵖ) (X : C) (s : (F.comp (CategoryTheory.yoneda.obj X)).sections) :

        The cone of F corresponding to an element in (F ⋙ yoneda.obj X).sections.

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          The yoneda embeddings jointly reflect limits.

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            A cocone is colimit iff it becomes limit after the application of yoneda.obj X for all X : C.

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              def CategoryTheory.Limits.coneOfSectionCompCoyoneda {C : Type u} [CategoryTheory.Category.{v, u} C] {J : Type w} [CategoryTheory.Category.{t, w} J] (F : CategoryTheory.Functor J C) (X : Cᵒᵖ) (s : (F.comp (CategoryTheory.coyoneda.obj X)).sections) :

              The cone of F corresponding to an element in (F ⋙ coyoneda.obj X).sections.

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                The coyoneda embeddings jointly reflect limits.

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                  A cone is limit iff it is so after the application of coyoneda.obj X for all X : Cᵒᵖ.

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                    The yoneda embedding yoneda.obj X : Cᵒᵖ ⥤ Type v for X : C preserves limits.

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                    The coyoneda embedding coyoneda.obj X : C ⥤ Type v for X : Cᵒᵖ preserves limits.

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                    • CategoryTheory.yonedaFunctorReflectsLimits = inferInstance
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                    • CategoryTheory.coyonedaFunctorReflectsLimits = inferInstance
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                    • F.corepresentablePreservesLimitsOfShape J = { preservesLimit := fun {K : CategoryTheory.Functor J C} => inferInstance }
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