Theorems · Definition · category theory

CategoryTheory.Limits.fiberwiseColimCompEvaluationIso

{C : Type u₁} →
  [inst : CategoryTheory.Category.{v₁, u₁} C] →
    {F : CategoryTheory.Functor C CategoryTheory.Cat} →
      {H : Type u₂} →
        [inst_1 : CategoryTheory.Category.{v₂, u₂} H] →
          [inst_2 : ∀ (c : C), CategoryTheory.Limits.HasColimitsOfShape (↑(F.obj c)) H] →
            (c : C) →
              (CategoryTheory.Limits.fiberwiseColim F H).comp ((CategoryTheory.evaluation C H).obj c) ≅
                ((CategoryTheory.Functor.whiskeringLeft (↑(F.obj c)) (CategoryTheory.Grothendieck F) H).obj
                      (CategoryTheory.Grothendieck.ι F c)).comp
                  CategoryTheory.Limits.colim

Composing fiberwiseColim F H with the evaluation functor (evaluation C H).obj c is naturally isomorphic to precomposing the Grothendieck inclusion Grothendieck.ι to colim.

Defined in

Mathlib.CategoryTheory.Limits.Shapes.Grothendieck

Cited by

4 results in Mathlib

Foundations

Depth 51 from the axioms · uses propext, Classical.choice, Quot.sound

Assumes

CategoryTheory.CategoryCategoryTheory.CategoryCategoryTheory.Limits.HasColimitsOfShape