Theorems · Theorem · category theory

CategoryTheory.Limits.Concrete.initial_of_empty_of_reflects

∀ {C : Type u} [inst : CategoryTheory.Category.{v, u} C] {FC : C → C → Type u_1} {CC : C → Type w}
  [inst_1 : (X Y : C) → FunLike (FC X Y) (CC X) (CC Y)] [inst_2 : CategoryTheory.ConcreteCategory C FC]
  [CategoryTheory.Limits.ReflectsColimit (CategoryTheory.Functor.empty C) (CategoryTheory.forget C)] (X : C),
  IsEmpty (CategoryTheory.ToType X) → Nonempty (CategoryTheory.Limits.IsInitial X)

If forget C reflects initials and ToType X is empty, then X is initial.

Defined in

Mathlib.CategoryTheory.Limits.Shapes.ConcreteCategory

Cited by

0 results in Mathlib

Foundations

Depth 39 from the axioms · uses propext, Classical.choice, Quot.sound

Assumes

CategoryTheory.CategoryFunLikeCategoryTheory.ConcreteCategoryCategoryTheory.Limits.ReflectsColimit