Theorems · Theorem · category theory
CategoryTheory.Limits.Types.initial_iff_empty
∀ (X : Type u), Nonempty (CategoryTheory.Limits.IsInitial X) ↔ IsEmpty X
An object in Type u is initial if and only if it is empty.
- Cited by
- 3 results in Mathlib
- Foundations
- Depth 38 from the axioms · uses propext, Classical.choice, Quot.sound
Around this declaration
Dashed lines are statement dependencies; solid lines are citations in proofs.
Cites10
Mathlib declarations this one mentions in its statement or cites explicitly in its proof. Plumbing is filtered out.
- DFunLike.coeproof · cited by 62,936
- CategoryTheory.ConcreteCategory.homproof · cited by 4,022
- IsEmptystatement and proof · cited by 759
- CategoryTheory.Limits.IsInitialstatement and proof · cited by 158
- CategoryTheory.Limits.IsInitial.toproof · cited by 119
- Equiv.toIsoproof · cited by 58
- Function.isEmptyproof · cited by 15
- CategoryTheory.Limits.IsInitial.ofIsoproof · cited by 15
- Equiv.equivOfIsEmptyproof · cited by 4
- CategoryTheory.Limits.Types.isInitialPEmptyproof · cited by 1
Cited by3
Results whose statement or proof uses this declaration.
- CategoryTheory.Limits.Concrete.initial_of_empty_of_reflectsproof · cited by 0
- CategoryTheory.Limits.Concrete.empty_of_initial_of_preservesproof · cited by 0