Theorems · Theorem · category theory

CategoryTheory.IsMonHom.normal_iff_normal_monoidHom

∀ {C : Type u_1} [inst : CategoryTheory.Category.{v_1, u_1} C] [inst_1 : CategoryTheory.CartesianMonoidalCategory C]
  {G H : C} [inst_2 : CategoryTheory.GrpObj G] [inst_3 : CategoryTheory.GrpObj H] {φ : H ⟶ G}
  [inst_4 : CategoryTheory.IsMonHom φ] [CategoryTheory.Mono φ],
  CategoryTheory.IsMonHom.Normal φ ↔ ∀ (X : C), (CategoryTheory.IsMonHom.monoidHom φ X).range.Normal

If φ is mono, it is a normal group homomorphism if and only if for all X the image of H(X) in G(X) is a normal subgroup.

Defined in

Mathlib.CategoryTheory.Monoidal.Cartesian.Normal

Cited by

0 results in Mathlib

Foundations

Depth 49 from the axioms · uses propext, Classical.choice, Quot.sound

Assumes

CategoryTheory.CategoryCategoryTheory.CartesianMonoidalCategoryCategoryTheory.GrpObjCategoryTheory.GrpObjCategoryTheory.IsMonHomCategoryTheory.Mono