Theorems · Theorem · category theory

CategoryTheory.MonoidalCategory.Arrow.PushoutProduct.hasLiftingProperty_mk_isInitial_iff

∀ {C : Type u} [inst : CategoryTheory.Category.{v, u} C] [inst_1 : CategoryTheory.Limits.HasPushouts C]
  [inst_2 : CategoryTheory.CartesianMonoidalCategory C] [inst_3 : CategoryTheory.MonoidalClosed C]
  [CategoryTheory.BraidedCategory C] {A B K L X Y : C} {g : K ⟶ L} {h : X ⟶ Y} (i : CategoryTheory.Limits.IsInitial A),
  CategoryTheory.HasLiftingProperty (CategoryTheory.Arrow.mk (i.to B) □ CategoryTheory.Arrow.mk g).hom h ↔
    CategoryTheory.HasLiftingProperty g ((CategoryTheory.ihom B).map h)

(∅ ⟶ B) □ g lifts against X ⟶ Y if and only if g lifts against B ⟹ X ⟶ B ⟹ Y.

Defined in

Mathlib.CategoryTheory.LiftingProperties.PushoutProduct

Cited by

2 results in Mathlib

Foundations

Depth 43 from the axioms · uses propext, Classical.choice, Quot.sound

Assumes

CategoryTheory.CategoryCategoryTheory.Limits.HasPushoutsCategoryTheory.CartesianMonoidalCategoryCategoryTheory.MonoidalClosedCategoryTheory.BraidedCategory