Theorems · Theorem · category theory

CategoryTheory.Limits.fiberwiseColimitLimitIso_hom_app

∀ {C : Type u₁} [inst : CategoryTheory.Category.{v₁, u₁} C] {H : Type u₂} [inst_1 : CategoryTheory.Category.{v₂, u₂} H]
  {J : Type u₃} [inst_2 : CategoryTheory.Category.{v₃, u₃} J] {F : CategoryTheory.Functor C CategoryTheory.Cat}
  (K : CategoryTheory.Functor J (CategoryTheory.Functor (CategoryTheory.Grothendieck F) H))
  [inst_3 : ∀ (c : C), CategoryTheory.Limits.HasColimitsOfShape (↑(F.obj c)) H]
  [inst_4 : CategoryTheory.Limits.HasLimitsOfShape J H]
  [inst_5 : ∀ (c : C), CategoryTheory.Limits.PreservesLimitsOfShape J CategoryTheory.Limits.colim] (X : C),
  (CategoryTheory.Limits.fiberwiseColimitLimitIso K).hom.app X =
    CategoryTheory.CategoryStruct.comp
      (CategoryTheory.Limits.HasColimit.isoOfNatIso
          (CategoryTheory.Limits.limitCompWhiskeringLeftIsoCompLimit K (CategoryTheory.Grothendieck.ι F X)).symm).hom
      (CategoryTheory.CategoryStruct.comp
        (CategoryTheory.preservesLimitIso CategoryTheory.Limits.colim
            (K.comp
              ((CategoryTheory.Functor.whiskeringLeft (↑(F.obj X)) (CategoryTheory.Grothendieck F) H).obj
                (CategoryTheory.Grothendieck.ι F X)))).hom
        (CategoryTheory.CategoryStruct.comp
          (CategoryTheory.Limits.HasLimit.isoOfNatIso
              (K.associator
                  ((CategoryTheory.Functor.whiskeringLeft (↑(F.obj X)) (CategoryTheory.Grothendieck F) H).obj
                    (CategoryTheory.Grothendieck.ι F X))
                  CategoryTheory.Limits.colim ≪≫
                K.isoWhiskerLeft (CategoryTheory.Limits.fiberwiseColimCompEvaluationIso X).symm ≪≫
                  (K.associator (CategoryTheory.Limits.fiberwiseColim F H)
                      ((CategoryTheory.evaluation C H).obj X)).symm)).hom
          (CategoryTheory.Limits.limitObjIsoLimitCompEvaluation (K.comp (CategoryTheory.Limits.fiberwiseColim F H))
              X).inv))

Defined in

Mathlib.CategoryTheory.Limits.Preserves.Grothendieck

Cited by

0 results in Mathlib

Foundations

Depth 54 from the axioms · uses propext, Classical.choice, Quot.sound

Assumes

CategoryTheory.CategoryCategoryTheory.CategoryCategoryTheory.CategoryCategoryTheory.Limits.HasColimitsOfShapeCategoryTheory.Limits.HasLimitsOfShapeCategoryTheory.Limits.PreservesLimitsOfShape