Theorems · Definition · category theory

CategoryTheory.Functor.toCostructuredArrow

{C : Type u₁} →
  [inst : CategoryTheory.Category.{v₁, u₁} C] →
    {D : Type u₂} →
      [inst_1 : CategoryTheory.Category.{v₂, u₂} D] →
        {E : Type u₃} →
          [inst_2 : CategoryTheory.Category.{v₃, u₃} E] →
            (G : CategoryTheory.Functor E C) →
              (F : CategoryTheory.Functor C D) →
                (X : D) →
                  (f : (Y : E) → F.obj (G.obj Y) ⟶ X) →
                    (∀ {Y Z : E} (g : Y ⟶ Z), CategoryTheory.CategoryStruct.comp (F.map (G.map g)) (f Z) = f Y) →
                      CategoryTheory.Functor E (CategoryTheory.CostructuredArrow F X)

Given F : C ⥤ D and X : D, to upgrade a functor G : E ⥤ C to a functor E ⥤ CostructuredArrow F X, it suffices to provide maps F.obj (G.obj Y) ⟶ X for all Y making the obvious triangles involving all F.map (G.map g) commute. This is of course the same as providing a cocone over F ⋙ G with cocone point X, see Functor.toCostructuredArrowIsoToCostructuredArrow.

Defined in

Mathlib.CategoryTheory.Comma.StructuredArrow.Basic

Cited by

4 results in Mathlib

Foundations

Depth 30 from the axioms · uses propext, Classical.choice, Quot.sound

Assumes

CategoryTheory.CategoryCategoryTheory.CategoryCategoryTheory.Category