Theorems · Theorem · category theory

CategoryTheory.Limits.CategoricalPullback.CatCommSqOver.transform_map_associator

∀ {A : Type u₁} {B : Type u₂} {C : Type u₃} [inst : CategoryTheory.Category.{v₁, u₁} A]
  [inst_1 : CategoryTheory.Category.{v₂, u₂} B] [inst_2 : CategoryTheory.Category.{v₃, u₃} C]
  {F : CategoryTheory.Functor A B} {G : CategoryTheory.Functor C B} {A₁ : Type u₄} {B₁ : Type u₅} {C₁ : Type u₆}
  [inst_3 : CategoryTheory.Category.{v₄, u₄} A₁] [inst_4 : CategoryTheory.Category.{v₅, u₅} B₁]
  [inst_5 : CategoryTheory.Category.{v₆, u₆} C₁] {F₁ : CategoryTheory.Functor A₁ B₁} {G₁ : CategoryTheory.Functor C₁ B₁}
  {A₂ : Type u₇} {B₂ : Type u₈} {C₂ : Type u₉} [inst_6 : CategoryTheory.Category.{v₇, u₇} A₂]
  [inst_7 : CategoryTheory.Category.{v₈, u₈} B₂] [inst_8 : CategoryTheory.Category.{v₉, u₉} C₂]
  {F₂ : CategoryTheory.Functor A₂ B₂} {G₂ : CategoryTheory.Functor C₂ B₂} {A₃ : Type u₁₀} {B₃ : Type u₁₁}
  {C₃ : Type u₁₂} [inst_9 : CategoryTheory.Category.{v₁₀, u₁₀} A₃] [inst_10 : CategoryTheory.Category.{v₁₁, u₁₁} B₃]
  [inst_11 : CategoryTheory.Category.{v₁₂, u₁₂} C₃] {F₃ : CategoryTheory.Functor A₃ B₃}
  {G₃ : CategoryTheory.Functor C₃ B₃} (X : Type u₁₃) [inst_12 : CategoryTheory.Category.{v₁₃, u₁₃} X]
  (ψ : CategoryTheory.Limits.CatCospanTransform F G F₁ G₁) (φ : CategoryTheory.Limits.CatCospanTransform F₁ G₁ F₂ G₂)
  (τ : CategoryTheory.Limits.CatCospanTransform F₂ G₂ F₃ G₃),
  (CategoryTheory.Limits.CategoricalPullback.CatCommSqOver.transform X).map (ψ.associator φ τ).hom =
    CategoryTheory.CategoryStruct.comp
      (CategoryTheory.Limits.CategoricalPullback.CatCommSqOver.transformObjComp X (ψ.comp φ) τ).hom
      (CategoryTheory.CategoryStruct.comp
        (CategoryTheory.Functor.whiskerRight
          (CategoryTheory.Limits.CategoricalPullback.CatCommSqOver.transformObjComp X ψ φ).hom
          ((CategoryTheory.Limits.CategoricalPullback.CatCommSqOver.transform X).obj τ))
        (CategoryTheory.CategoryStruct.comp
          (((CategoryTheory.Limits.CategoricalPullback.CatCommSqOver.transform X).obj ψ).associator
              ((CategoryTheory.Limits.CategoricalPullback.CatCommSqOver.transform X).obj φ)
              ((CategoryTheory.Limits.CategoricalPullback.CatCommSqOver.transform X).obj τ)).hom
          (CategoryTheory.CategoryStruct.comp
            (((CategoryTheory.Limits.CategoricalPullback.CatCommSqOver.transform X).obj ψ).whiskerLeft
              (CategoryTheory.Limits.CategoricalPullback.CatCommSqOver.transformObjComp X φ τ).inv)
            (CategoryTheory.Limits.CategoricalPullback.CatCommSqOver.transformObjComp X ψ (φ.comp τ)).inv)))

Defined in

Mathlib.CategoryTheory.Limits.Shapes.Pullback.Categorical.Basic

Cited by

0 results in Mathlib

Foundations

Depth 43 from the axioms · uses propext, Classical.choice, Quot.sound

Assumes

CategoryTheory.CategoryCategoryTheory.CategoryCategoryTheory.CategoryCategoryTheory.CategoryCategoryTheory.CategoryCategoryTheory.CategoryCategoryTheory.CategoryCategoryTheory.CategoryCategoryTheory.CategoryCategoryTheory.CategoryCategoryTheory.CategoryCategoryTheory.CategoryCategoryTheory.Category