Theorems · Definition · category theory

CategoryTheory.GrothendieckTopology.Point.sheafFiberComapIso

{C : Type u_1} →
  {D : Type u_2} →
    [inst : CategoryTheory.Category.{v_1, u_1} C] →
      [inst_1 : CategoryTheory.Category.{v_2, u_2} D] →
        {K : CategoryTheory.GrothendieckTopology D} →
          (Φ : K.Point) →
            (F : CategoryTheory.Functor C D) →
              [inst_2 : CategoryTheory.RepresentablyFlat F] →
                {J : CategoryTheory.GrothendieckTopology C} →
                  (hF : CategoryTheory.CoverPreserving J K F) →
                    [inst_3 : CategoryTheory.InitiallySmall (F.comp Φ.fiber).Elements] →
                      (A : Type u_3) →
                        [inst_4 : CategoryTheory.Category.{v, u_3} A] →
                          [CategoryTheory.Limits.HasProducts A] →
                            [inst_6 : F.IsContinuous J K] →
                              [inst_7 : (F.sheafPushforwardContinuous A J K).IsRightAdjoint] →
                                [inst_8 : CategoryTheory.Limits.HasColimitsOfSize.{w, w, v, u_3} A] →
                                  (Φ.comap F hF).sheafFiber ≅ (F.sheafPullback A J K).comp Φ.sheafFiber

Given a continuous functor F : C ⥤ D between sites (C, J) and (D, K), and a point Φ of (D, K), the fiber functor on sheaves of the point Φ.comap F hF on (C, J) identifies to the composition F.sheafPullback A J K ⋙ Φ.sheafFiber.

Defined in

Mathlib.CategoryTheory.Sites.Point.Comap

Cited by

2 results in Mathlib

Foundations

Depth 73 from the axioms · uses propext, Classical.choice, Quot.sound

Assumes

CategoryTheory.CategoryCategoryTheory.CategoryCategoryTheory.RepresentablyFlatCategoryTheory.InitiallySmallCategoryTheory.CategoryCategoryTheory.Limits.HasProductsCategoryTheory.Functor.IsContinuousCategoryTheory.Functor.IsRightAdjointCategoryTheory.Limits.HasColimitsOfSize