Theorems · Definition · category theory

CategoryTheory.Limits.PullbackCone.isLimitMkIdId

{C : Type u} →
  [inst : CategoryTheory.Category.{v, u} C] →
    {X Y : C} →
      (f : X ⟶ Y) →
        [CategoryTheory.Mono f] →
          CategoryTheory.Limits.IsLimit
            (CategoryTheory.Limits.PullbackCone.mk (CategoryTheory.CategoryStruct.id X)
              (CategoryTheory.CategoryStruct.id X) ⋯)

The pullback cone (𝟙 X, 𝟙 X) for the pair (f, f) is a limit if f is a mono. The converse is shown in mono_of_pullback_is_id.

Defined in

Mathlib.CategoryTheory.Limits.Shapes.Pullback.Mono

Cited by

4 results in Mathlib

Foundations

Depth 28 from the axioms · uses propext, Classical.choice, Quot.sound

Assumes

CategoryTheory.CategoryCategoryTheory.Mono