Theorems · Theorem · category theory

CategoryTheory.ObjectProperty.IsConservativeFamilyOfPoints.isMonoidal_W

∀ {C : Type u} [inst : CategoryTheory.Category.{v, u} C] [CategoryTheory.LocallySmall.{w, v, u} C]
  {J : CategoryTheory.GrothendieckTopology C} {P : CategoryTheory.ObjectProperty J.Point},
  P.IsConservativeFamilyOfPoints →
    ∀ (A : Type u') [inst_2 : CategoryTheory.Category.{v', u'} A] [inst_3 : CategoryTheory.MonoidalCategory A]
      [CategoryTheory.Limits.HasColimitsOfSize.{w, w, v', u'} A] [CategoryTheory.Limits.HasProducts A]
      {FC : A → A → Type u_1} {CC : A → Type w} [inst_6 : (X Y : A) → FunLike (FC X Y) (CC X) (CC Y)]
      [inst_7 : CategoryTheory.ConcreteCategory A FC] [CategoryTheory.HasWeakSheafify J A]
      [(CategoryTheory.forget A).ReflectsIsomorphisms]
      [CategoryTheory.Limits.PreservesFilteredColimitsOfSize.{w, w, v', w, u', w + 1} (CategoryTheory.forget A)]
      [∀ (X : A),
          CategoryTheory.Limits.PreservesFilteredColimitsOfSize.{w, w, v', v', u', u'}
            (CategoryTheory.MonoidalCategory.tensorLeft X)]
      [∀ (X : A),
          CategoryTheory.Limits.PreservesFilteredColimitsOfSize.{w, w, v', v', u', u'}
            (CategoryTheory.MonoidalCategory.tensorRight X)]
      [J.HasSheafCompose (CategoryTheory.forget A)], J.W.IsMonoidal

Defined in

Mathlib.CategoryTheory.Sites.Point.IsMonoidalW

Cited by

0 results in Mathlib

Foundations

Depth 64 from the axioms · uses propext, Classical.choice, Quot.sound

Assumes

CategoryTheory.CategoryCategoryTheory.LocallySmallCategoryTheory.CategoryCategoryTheory.MonoidalCategoryCategoryTheory.Limits.HasColimitsOfSizeCategoryTheory.Limits.HasProductsFunLikeCategoryTheory.ConcreteCategoryCategoryTheory.HasWeakSheafifyCategoryTheory.Functor.ReflectsIsomorphismsCategoryTheory.Limits.PreservesFilteredColimitsOfSizeCategoryTheory.Limits.PreservesFilteredColimitsOfSizeCategoryTheory.Limits.PreservesFilteredColimitsOfSizeCategoryTheory.GrothendieckTopology.HasSheafCompose