Theorems · Theorem · category theory

CategoryTheory.ModObj.mul_smul

∀ {C : Type u₁} {inst : CategoryTheory.Category.{v₁, u₁} C} {inst_1 : CategoryTheory.MonoidalCategory C} {D : Type u₂}
  {inst_2 : CategoryTheory.Category.{v₂, u₂} D} {inst_3 : CategoryTheory.MonoidalCategory.MonoidalLeftAction C D}
  {M : C} {inst_4 : CategoryTheory.MonObj M} (X : D) [self : CategoryTheory.ModObj M X],
  CategoryTheory.CategoryStruct.comp
      (CategoryTheory.MonoidalCategory.MonoidalLeftActionStruct.actionHomLeft CategoryTheory.MonObj.mul X)
      CategoryTheory.ModObj.smul =
    CategoryTheory.CategoryStruct.comp
      (CategoryTheory.MonoidalCategory.MonoidalLeftActionStruct.actionAssocIso M M X).hom
      (CategoryTheory.CategoryStruct.comp
        (CategoryTheory.MonoidalCategory.MonoidalLeftActionStruct.actionHomRight M CategoryTheory.ModObj.smul)
        CategoryTheory.ModObj.smul)

The action map is compatible with multiplication.

Defined in

Mathlib.CategoryTheory.Monoidal.Mod

Cited by

4 results in Mathlib

Foundations

Depth 5 from the axioms · uses no axioms

Assumes

CategoryTheory.ModObj