Theorems · Theorem · category theory

CategoryTheory.GrothendieckTopology.liftToPlusObjLimitObj_fac

∀ {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]
  [inst_6 : ∀ (X : C), CategoryTheory.Limits.PreservesColimitsOfShape (J.Cover X)ᵒᵖ (CategoryTheory.forget D)]
  [inst_7 : ∀ (X : C), Small.{t, max u v} (J.Cover X)ᵒᵖ] {K : Type s} [inst_8 : CategoryTheory.SmallCategory K]
  [inst_9 : CategoryTheory.FinCategory K] [inst_10 : CategoryTheory.Limits.HasLimitsOfShape K D]
  [inst_11 : CategoryTheory.Limits.PreservesLimitsOfShape K (CategoryTheory.forget D)]
  [inst_12 : 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)))))
  (k : K),
  CategoryTheory.CategoryStruct.comp (CategoryTheory.GrothendieckTopology.liftToPlusObjLimitObj F X S)
      ((J.plusMap (CategoryTheory.Limits.limit.π F k)).app (Opposite.op X)) =
    S.π.app k

Defined in

Mathlib.CategoryTheory.Sites.LeftExact

Cited by

0 results in Mathlib

Foundations

Depth 83 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