Theorems · Definition · category theory

CategoryTheory.GrothendieckTopology.liftToPlusObjLimitObj

{C : Type u} →
  [inst : CategoryTheory.Category.{v, u} C] →
    {J : CategoryTheory.GrothendieckTopology C} →
      {D : Type w} →
        [inst_1 : CategoryTheory.Category.{t, w} D] →
          [inst_2 :
              ∀ (P : CategoryTheory.Functor Cᵒᵖ D) (X : C) (S : J.Cover X),
                CategoryTheory.Limits.HasMultiequalizer (S.index P)] →
            [inst_3 : ∀ (X : C), CategoryTheory.Limits.HasColimitsOfShape (J.Cover X)ᵒᵖ D] →
              {FD : D → D → Type u_1} →
                {CD : D → Type t} →
                  [inst_4 : (X Y : D) → FunLike (FD X Y) (CD X) (CD Y)] →
                    [inst_5 : CategoryTheory.ConcreteCategory D FD] →
                      [∀ (X : C),
                            CategoryTheory.Limits.PreservesColimitsOfShape (J.Cover X)ᵒᵖ (CategoryTheory.forget D)] →
                        [∀ (X : C), Small.{t, max u v} (J.Cover X)ᵒᵖ] →
                          {K : Type s} →
                            [inst_8 : CategoryTheory.SmallCategory K] →
                              [CategoryTheory.FinCategory K] →
                                [inst_10 : CategoryTheory.Limits.HasLimitsOfShape K D] →
                                  [CategoryTheory.Limits.PreservesLimitsOfShape K (CategoryTheory.forget D)] →
                                    [CategoryTheory.Limits.ReflectsLimitsOfShape K (CategoryTheory.forget D)] →
                                      (F : CategoryTheory.Functor K (CategoryTheory.Functor Cᵒᵖ D)) →
                                        (X : C) →
                                          (S :
                                              CategoryTheory.Limits.Cone
                                                (F.comp
                                                  ((J.plusFunctor D).comp
                                                    ((CategoryTheory.evaluation Cᵒᵖ D).obj (Opposite.op X))))) →
                                            S.pt ⟶ (J.plusObj (CategoryTheory.Limits.limit F)).obj (Opposite.op X)

An auxiliary definition to be used in the proof that J.plusFunctor D commutes with finite limits.

Defined in

Mathlib.CategoryTheory.Sites.LeftExact

Cited by

1 results in Mathlib

Foundations

Depth 82 from the axioms · uses propext, Classical.choice, Quot.sound

Assumes

CategoryTheory.CategoryCategoryTheory.CategoryCategoryTheory.Limits.HasMultiequalizerCategoryTheory.Limits.HasColimitsOfShapeFunLikeCategoryTheory.ConcreteCategoryCategoryTheory.Limits.PreservesColimitsOfShapeSmallCategoryTheory.SmallCategoryCategoryTheory.FinCategoryCategoryTheory.Limits.HasLimitsOfShapeCategoryTheory.Limits.PreservesLimitsOfShapeCategoryTheory.Limits.ReflectsLimitsOfShape