Theorems · Definition · category theory

CategoryTheory.Limits.CategoricalPullback.CatCommSqOver.transformObjComp

{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} →
                                                (X : Type u₁₀) →
                                                  [inst_9 : CategoryTheory.Category.{v₁₀, u₁₀} X] →
                                                    (ψ : CategoryTheory.Limits.CatCospanTransform F G F₁ G₁) →
                                                      (ψ' : CategoryTheory.Limits.CatCospanTransform F₁ G₁ F₂ G₂) →
                                                        (CategoryTheory.Limits.CategoricalPullback.CatCommSqOver.transform
                                                                X).obj
                                                            (ψ.comp ψ') ≅
                                                          ((CategoryTheory.Limits.CategoricalPullback.CatCommSqOver.transform
                                                                    X).obj
                                                                ψ).comp
                                                            ((CategoryTheory.Limits.CategoricalPullback.CatCommSqOver.transform
                                                                  X).obj
                                                              ψ')

The construction CatCommSqOver.transform respects vertical composition of CatCospanTransforms.

Defined in

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

Cited by

10 results in Mathlib

Foundations

Depth 42 from the axioms · uses propext, Classical.choice, Quot.sound

Assumes

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