Theorems · Theorem · category theory

CategoryTheory.Limits.fiberwiseColimitLimitIso_inv_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).inv.app X =
    CategoryTheory.CategoryStruct.comp
      (CategoryTheory.Limits.limitObjIsoLimitCompEvaluation (K.comp (CategoryTheory.Limits.fiberwiseColim F H)) 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)).inv
        (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)))).inv
          (CategoryTheory.Limits.HasColimit.isoOfNatIso
              (CategoryTheory.Limits.limitCompWhiskeringLeftIsoCompLimit K
                  (CategoryTheory.Grothendieck.ι F X)).symm).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