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Mathlib.CategoryTheory.LiftingProperties.Over

Lifting properties in Over categories #

In this file, we show that if sq is a commutative square in a category Over S for S : C, there is a lift for sq if there is a lift for the underlying commutative square in the category C. It follows that if i and p are morphisms in Over S, then i has the left lifting property with respect to p when i.left has the left lifting property with respect to p.left.

theorem CategoryTheory.CommSq.HasLift.over {C : Type u} [Category.{v, u} C] {S : C} {X₁ X₂ X₃ X₄ : Over S} {t : X₁ X₂} {l : X₁ X₃} {r : X₂ X₄} {b : X₃ X₄} {sq : CommSq t l r b} [.HasLift] :
theorem CategoryTheory.HasLiftingProperty.over {C : Type u} [Category.{v, u} C] {S : C} {A B X Y : Over S} (i : A B) (p : X Y) [HasLiftingProperty i.left p.left] :