Theorems · Definition · category theory

CategoryTheory.MorphismProperty.Over.pullbackComp

{T : Type u_1} →
  [inst : CategoryTheory.Category.{v_1, u_1} T] →
    {P Q : CategoryTheory.MorphismProperty T} →
      [inst_1 : Q.IsMultiplicative] →
        {X Y Z : T} →
          (f : X ⟶ Y) →
            (g : Y ⟶ Z) →
              [inst_2 : P.IsStableUnderBaseChangeAlong f] →
                [inst_3 : P.IsStableUnderBaseChangeAlong g] →
                  [inst_4 : P.HasPullbacksAlong f] →
                    [inst_5 : P.HasPullbacksAlong g] →
                      [Q.RespectsIso] →
                        [inst_7 : Q.IsStableUnderBaseChange] →
                          (fg : optParam (X ⟶ Z) (CategoryTheory.CategoryStruct.comp f g)) →
                            (hfg :
                                autoParam (fg = CategoryTheory.CategoryStruct.comp f g)
                                  CategoryTheory.MorphismProperty.Over.pullbackComp._auto_1) →
                              CategoryTheory.MorphismProperty.Over.pullback P Q fg ≅
                                (CategoryTheory.MorphismProperty.Over.pullback P Q g).comp
                                  (CategoryTheory.MorphismProperty.Over.pullback P Q f)

Over.pullback commutes with composition.

Defined in

Mathlib.CategoryTheory.MorphismProperty.OverAdjunction

Cited by

5 results in Mathlib

Foundations

Depth 71 from the axioms · uses propext, Classical.choice, Quot.sound

Assumes

CategoryTheory.CategoryCategoryTheory.MorphismProperty.IsMultiplicativeCategoryTheory.MorphismProperty.IsStableUnderBaseChangeAlongCategoryTheory.MorphismProperty.IsStableUnderBaseChangeAlongCategoryTheory.MorphismProperty.HasPullbacksAlongCategoryTheory.MorphismProperty.HasPullbacksAlongCategoryTheory.MorphismProperty.RespectsIsoCategoryTheory.MorphismProperty.IsStableUnderBaseChange