Theorems · Theorem · category theory
CategoryTheory.PreGaloisCategory.not_initial_iff_fiber_nonempty
∀ {C : Type u₁} [inst : CategoryTheory.Category.{u₂, u₁} C] (F : CategoryTheory.Functor C FintypeCat)
[inst_1 : CategoryTheory.PreGaloisCategory C] [CategoryTheory.PreGaloisCategory.FiberFunctor F] (X : C),
(∀ (a : CategoryTheory.Limits.IsInitial X), False) ↔ Nonempty (F.obj X).objAn object is not initial if and only if its fiber is nonempty.
- Defined in
- Mathlib.CategoryTheory.Galois.Basic
- Cited by
- 2 results in Mathlib
- Foundations
- Depth 76 from the axioms · uses propext, Classical.choice, Quot.sound
Around this declaration
Dashed lines are statement dependencies; solid lines are citations in proofs.
Cites12
Mathlib declarations this one mentions in its statement or cites explicitly in its proof. Plumbing is filtered out.
- CategoryTheory.Categorystatement and proof · cited by 32,673
- CategoryTheory.Functor.objstatement and proof · cited by 19,642
- CategoryTheory.Functorstatement and proof · cited by 16,252
- Finitestatement · cited by 3,029
- CategoryTheory.ObjectProperty.FullSubcategory.objstatement and proof · cited by 1,316
- IsEmptyproof · cited by 759
- FintypeCatstatement and proof · cited by 217
- CategoryTheory.Limits.IsInitialstatement and proof · cited by 158
- CategoryTheory.PreGaloisCategory.FiberFunctorstatement and proof · cited by 66
- CategoryTheory.PreGaloisCategorystatement and proof · cited by 21
- not_isEmpty_iffproof · cited by 19
- CategoryTheory.PreGaloisCategory.initial_iff_fiber_emptyproof · cited by 2
Cited by2
Results whose statement or proof uses this declaration.
- CategoryTheory.FintypeCat.Action.isConnected_of_transitiveproof · cited by 1
- CategoryTheory.FintypeCat.Action.pretransitive_of_isConnectedproof · cited by 1