Theorems · Theorem · category theory

CategoryTheory.preserves_fin_of_preserves_binary_and_initial

∀ {C : Type u} [inst : CategoryTheory.Category.{v, u} C] {D : Type u'} [inst_1 : CategoryTheory.Category.{v', u'} D]
  (F : CategoryTheory.Functor C D)
  [CategoryTheory.Limits.PreservesColimitsOfShape (CategoryTheory.Discrete CategoryTheory.Limits.WalkingPair) F]
  [CategoryTheory.Limits.PreservesColimitsOfShape (CategoryTheory.Discrete PEmpty.{1}) F]
  [CategoryTheory.Limits.HasFiniteCoproducts C] (n : ℕ) (f : Fin n → C),
  CategoryTheory.Limits.PreservesColimit (CategoryTheory.Discrete.functor f) F

If F preserves the initial object and binary coproducts, then it preserves products indexed by Fin n for any n.

Defined in

Mathlib.CategoryTheory.Limits.Constructions.FiniteProductsOfBinaryProducts

Cited by

1 results in Mathlib

Foundations

Depth 63 from the axioms · uses propext, Classical.choice, Quot.sound

Assumes

CategoryTheory.CategoryCategoryTheory.CategoryCategoryTheory.Limits.PreservesColimitsOfShapeCategoryTheory.Limits.PreservesColimitsOfShapeCategoryTheory.Limits.HasFiniteCoproducts