Theorems · Definition · category theory

CategoryTheory.GrpObj.commutator

{C : Type u} →
  [inst : CategoryTheory.Category.{v, u} C] →
    [inst_1 : CategoryTheory.CartesianMonoidalCategory C] →
      (G : C) → [CategoryTheory.GrpObj G] → CategoryTheory.MonoidalCategoryStruct.tensorObj G G ⟶ G

The commutator of G as a morphism. This is the map (x, y) ↦ x * y * x⁻¹ * y⁻¹, see CategoryTheory.GrpObj.lift_commutator_eq_mul_mul_inv_inv. This morphism is constant with value 1 if and only if G is commutative (see CategoryTheory.isCommMonObj_iff_commutator_eq_toUnit_η).

Defined in

Mathlib.CategoryTheory.Monoidal.Cartesian.Grp

Cited by

9 results in Mathlib

Foundations

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

Assumes

CategoryTheory.CategoryCategoryTheory.CartesianMonoidalCategoryCategoryTheory.GrpObj