Theorems · Theorem · category theory

CategoryTheory.IsPullback.of_over_iso

∀ {C : Type u₁} [inst : CategoryTheory.Category.{v₁, u₁} C] {X Y Z : C} {f : X ⟶ Z} {g : Y ⟶ Z}
  [inst_1 : CategoryTheory.Limits.HasPullbacksAlong g] {P : C} {p : P ⟶ X}
  (e : CategoryTheory.Over.mk p ≅ CategoryTheory.Over.mk (CategoryTheory.Limits.pullback.fst f g)),
  CategoryTheory.IsPullback p
    (CategoryTheory.CategoryStruct.comp (CategoryTheory.Over.Hom.left e.hom) (CategoryTheory.Limits.pullback.snd f g)) f
    g

An isomorphism Over.mk p ≅ Over.mk (pullback.fst f g) in Over X yields an IsPullback square.

Defined in

Mathlib.CategoryTheory.Limits.Shapes.Pullback.IsPullback.Basic

Cited by

1 results in Mathlib

Foundations

Depth 38 from the axioms · uses propext, Classical.choice, Quot.sound

Assumes

CategoryTheory.CategoryCategoryTheory.Limits.HasPullbacksAlong