Theorems · Definition · category theory

CategoryTheory.Limits.isLimitOfHasPullbackOfPreservesLimit

{C : Type u₁} →
  [inst : CategoryTheory.Category.{v₁, u₁} C] →
    {D : Type u₂} →
      [inst_1 : CategoryTheory.Category.{v₂, u₂} D] →
        (G : CategoryTheory.Functor C D) →
          {X Y Z : C} →
            (f : X ⟶ Z) →
              (g : Y ⟶ Z) →
                [CategoryTheory.Limits.PreservesLimit (CategoryTheory.Limits.cospan f g) G] →
                  [inst_3 : CategoryTheory.Limits.HasPullback f g] →
                    have this := ⋯;
                    CategoryTheory.Limits.IsLimit
                      (CategoryTheory.Limits.PullbackCone.mk (G.map (CategoryTheory.Limits.pullback.fst f g))
                        (G.map (CategoryTheory.Limits.pullback.snd f g)) this)

If G preserves pullbacks and C has them, then the pullback cone constructed of the mapped morphisms of the pullback cone is a limit.

Defined in

Mathlib.CategoryTheory.Limits.Preserves.Shapes.Pullbacks

Cited by

7 results in Mathlib

Foundations

Depth 37 from the axioms · uses propext, Classical.choice, Quot.sound

Assumes

CategoryTheory.CategoryCategoryTheory.CategoryCategoryTheory.Limits.PreservesLimitCategoryTheory.Limits.HasPullback