Theorems · Theorem · category theory

CategoryTheory.Localization.preservesProductsOfShape

∀ {C : Type u₁} {D : Type u₂} [inst : CategoryTheory.Category.{v₁, u₁} C] [inst_1 : CategoryTheory.Category.{v₂, u₂} D]
  (L : CategoryTheory.Functor C D) (W : CategoryTheory.MorphismProperty C) [L.IsLocalization W] [W.ContainsIdentities]
  (J : Type) [Finite J] [CategoryTheory.Limits.HasProductsOfShape J C] [W.IsStableUnderProductsOfShape J],
  CategoryTheory.Limits.PreservesLimitsOfShape (CategoryTheory.Discrete J) L

When C has finite products indexed by J, W : MorphismProperty C contains identities and is stable under products indexed by J, then any localization functor for W preserves finite products indexed by J.

Defined in

Mathlib.CategoryTheory.Localization.FiniteProducts

Cited by

1 results in Mathlib

Foundations

Depth 96 from the axioms · uses propext, Classical.choice, Quot.sound

Assumes

CategoryTheory.CategoryCategoryTheory.CategoryCategoryTheory.Functor.IsLocalizationCategoryTheory.MorphismProperty.ContainsIdentitiesFiniteCategoryTheory.Limits.HasProductsOfShapeCategoryTheory.MorphismProperty.IsStableUnderProductsOfShape