Theorems · Definition · category theory

CategoryTheory.Functor.relativelyRepresentable.lift

{C : Type u₁} →
  [inst : CategoryTheory.Category.{v₁, u₁} C] →
    {D : Type u₂} →
      [inst_1 : CategoryTheory.Category.{v₂, u₂} D] →
        {F : CategoryTheory.Functor C D} →
          {X Y : D} →
            {f : X ⟶ Y} →
              (hf : F.relativelyRepresentable f) →
                {a : C} →
                  {g : F.obj a ⟶ Y} →
                    {c : C} →
                      (i : F.obj c ⟶ X) →
                        (h : c ⟶ a) →
                          CategoryTheory.CategoryStruct.comp i f = CategoryTheory.CategoryStruct.comp (F.map h) g →
                            [F.Full] → c ⟶ hf.pullback g

The lift (in C) obtained from the universal property of F.obj (hf.pullback g), in the case when the cone point is in the image of F.obj.

Defined in

Mathlib.CategoryTheory.MorphismProperty.Representable

Cited by

7 results in Mathlib

Foundations

Depth 27 from the axioms · uses propext, Classical.choice, Quot.sound

Assumes

CategoryTheory.CategoryCategoryTheory.CategoryCategoryTheory.Functor.Full