Theorems · Theorem · category theory

CategoryTheory.GrothendieckTopology.Point.sheafFiberCompIso_hom_app

∀ {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)
  [inst_5 : CategoryTheory.LocallySmall.{w, v, u} C]
  [inst_6 : CategoryTheory.Limits.PreservesFilteredColimitsOfSize.{w, w, v', v'', u', u''} F]
  [inst_7 : J.HasSheafCompose F] (X : CategoryTheory.Sheaf J A),
  (Φ.sheafFiberCompIso F).hom.app X =
    CategoryTheory.CategoryStruct.comp ((Φ.presheafFiberCompIso F).hom.app X.obj)
      (CategoryTheory.CategoryStruct.id (F.obj (Φ.presheafFiber.obj X.obj)))

Defined in

Mathlib.CategoryTheory.Sites.Point.Basic

Cited by

0 results in Mathlib

Foundations

Depth 58 from the axioms · uses propext, Classical.choice, Quot.sound

Assumes

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