Theorems · Theorem · category theory

CategoryTheory.PreGaloisCategory.fiber_in_connected_component

∀ {C : Type u₁} [inst : CategoryTheory.Category.{u₂, u₁} C] [inst_1 : CategoryTheory.GaloisCategory C]
  (F : CategoryTheory.Functor C FintypeCat) [CategoryTheory.PreGaloisCategory.FiberFunctor F] (X : C)
  (x : (F.obj X).obj),
  ∃ Y i y,
    (CategoryTheory.ConcreteCategory.hom (F.map i)) y = x ∧
      CategoryTheory.PreGaloisCategory.IsConnected Y ∧ CategoryTheory.Mono i

Every element in the fiber of X lies in some connected component of X.

Defined in

Mathlib.CategoryTheory.Galois.Decomposition

Cited by

2 results in Mathlib

Foundations

Depth 98 from the axioms · uses propext, Classical.choice, Quot.sound

Assumes

CategoryTheory.CategoryCategoryTheory.GaloisCategoryCategoryTheory.PreGaloisCategory.FiberFunctor