Theorems · Definition · category theory

CategoryTheory.GrothendieckTopology.Point.sheafFiberCompIso

{C : Type u} →
  [inst : CategoryTheory.Category.{v, u} C] →
    {J : CategoryTheory.GrothendieckTopology C} →
      (Φ : J.Point) →
        {A : Type u'} →
          [inst_1 : CategoryTheory.Category.{v', u'} A] →
            {B : Type u''} →
              [inst_2 : CategoryTheory.Category.{v'', u''} B] →
                [inst_3 : CategoryTheory.Limits.HasColimitsOfSize.{w, w, v', u'} A] →
                  [inst_4 : CategoryTheory.Limits.HasColimitsOfSize.{w, w, v'', u''} B] →
                    (F : CategoryTheory.Functor A B) →
                      [CategoryTheory.LocallySmall.{w, v, u} C] →
                        [CategoryTheory.Limits.PreservesFilteredColimitsOfSize.{w, w, v', v'', u', u''} F] →
                          [inst_7 : J.HasSheafCompose F] →
                            (CategoryTheory.sheafCompose J F).comp Φ.sheafFiber ≅ Φ.sheafFiber.comp F

If Φ is a point of a site and F : A ⥤ B is a functor which preserves filtered colimits, then taking fibers of sheaves at Φ commutes with F.

Defined in

Mathlib.CategoryTheory.Sites.Point.Basic

Cited by

3 results in Mathlib

Foundations

Depth 57 from the axioms · uses propext, Classical.choice, Quot.sound

Assumes

CategoryTheory.CategoryCategoryTheory.CategoryCategoryTheory.CategoryCategoryTheory.Limits.HasColimitsOfSizeCategoryTheory.Limits.HasColimitsOfSizeCategoryTheory.LocallySmallCategoryTheory.Limits.PreservesFilteredColimitsOfSizeCategoryTheory.GrothendieckTopology.HasSheafCompose