Theorems · Theorem · category theory

CategoryTheory.eHom_whisker_cancel

∀ (V : Type u') [inst : CategoryTheory.Category.{v', u'} V] [inst_1 : CategoryTheory.MonoidalCategory V] {C : Type u}
  [inst_2 : CategoryTheory.Category.{v, u} C] [inst_3 : CategoryTheory.EnrichedOrdinaryCategory V C] {X Y Y₁ Z : C}
  (α : Y ≅ Y₁),
  CategoryTheory.CategoryStruct.comp
      (CategoryTheory.MonoidalCategoryStruct.whiskerRight (CategoryTheory.eHomWhiskerLeft V X α.hom) (Y ⟶[V] Z))
      (CategoryTheory.CategoryStruct.comp
        (CategoryTheory.MonoidalCategoryStruct.whiskerLeft (X ⟶[V] Y₁) (CategoryTheory.eHomWhiskerRight V α.inv Z))
        (CategoryTheory.eComp V X Y₁ Z)) =
    CategoryTheory.eComp V X Y Z

Given an isomorphism α : Y ≅ Y₁ in C, the enriched composition map eComp V X Y Z : (X ⟶[V] Y) ⊗ (Y ⟶[V] Z) ⟶ (X ⟶[V] Z) factors through the V object (X ⟶[V] Y₁) ⊗ (Y₁ ⟶[V] Z) via the map defined by whiskering in the middle with α.hom and α.inv.

Defined in

Mathlib.CategoryTheory.Enriched.Ordinary.Basic

Cited by

2 results in Mathlib

Foundations

Depth 26 from the axioms · uses propext, Classical.choice, Quot.sound

Assumes

CategoryTheory.CategoryCategoryTheory.MonoidalCategoryCategoryTheory.CategoryCategoryTheory.EnrichedOrdinaryCategory