Theorems · Theorem · category theory

preservesBinaryCoproducts_of_preservesInitial_and_pushouts

∀ {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.HasInitial C] [CategoryTheory.Limits.HasPushouts C]
  [CategoryTheory.Limits.PreservesColimitsOfShape (CategoryTheory.Discrete PEmpty.{1}) F]
  [CategoryTheory.Limits.PreservesColimitsOfShape CategoryTheory.Limits.WalkingSpan F],
  CategoryTheory.Limits.PreservesColimitsOfShape (CategoryTheory.Discrete CategoryTheory.Limits.WalkingPair) F

A functor that preserves initial objects and pushouts preserves binary coproducts.

Defined in

Mathlib.CategoryTheory.Limits.Constructions.BinaryProducts

Cited by

1 results in Mathlib

Foundations

Depth 39 from the axioms · uses propext, Classical.choice, Quot.sound

Assumes

CategoryTheory.CategoryCategoryTheory.CategoryCategoryTheory.Limits.HasInitialCategoryTheory.Limits.HasPushoutsCategoryTheory.Limits.PreservesColimitsOfShapeCategoryTheory.Limits.PreservesColimitsOfShape