Theorems · Definition · category theory

CategoryTheory.MorphismProperty.overEquivOfIsInitial

{C : Type u_1} →
  [inst : CategoryTheory.Category.{v_1, u_1} C] →
    (P Q : CategoryTheory.MorphismProperty C) →
      [P.ContainsIdentities] →
        [inst_2 : Q.IsMultiplicative] →
          [Q.RespectsIso] →
            [CategoryTheory.Limits.HasStrictInitialObjects C] →
              (X : C) → CategoryTheory.Limits.IsInitial X → (P.Over Q X ≌ CategoryTheory.Discrete PUnit.{w + 1})

If C has strict initial objects and X is an initial object, the category P.Over Q X is equivalent to a point.

Defined in

Mathlib.CategoryTheory.Comma.Over.StrictInitial

Cited by

4 results in Mathlib

Foundations

Depth 67 from the axioms · uses propext, Classical.choice, Quot.sound

Assumes

CategoryTheory.CategoryCategoryTheory.MorphismProperty.ContainsIdentitiesCategoryTheory.MorphismProperty.IsMultiplicativeCategoryTheory.MorphismProperty.RespectsIsoCategoryTheory.Limits.HasStrictInitialObjects