Theorems · Definition · category theory

CategoryTheory.AddGrpObj.addCommutator

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

The commutator of G as a morphism. This is the map (x, y) ↦ x + y + (-x) + (-y), see CategoryTheory.AddGrpObj.lift_commutator_eq_add_add_neg_neg. This morphism is constant with value 0 if and only if G is commutative (see CategoryTheory.isCommAddMonObj_iff_commutator_eq_toAddUnit_η).

Defined in

Mathlib.CategoryTheory.Monoidal.Cartesian.Grp

Cited by

7 results in Mathlib

Foundations

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

Assumes

CategoryTheory.CategoryCategoryTheory.CartesianMonoidalCategoryCategoryTheory.AddGrpObj