Theorems · Definition · category theory

CategoryTheory.SingleFunctors.lift

{C : Type u_1} →
  {D : Type u_2} →
    {E : Type u_3} →
      [inst : CategoryTheory.Category.{u_5, u_1} C] →
        [inst_1 : CategoryTheory.Category.{u_6, u_2} D] →
          [inst_2 : CategoryTheory.Category.{u_7, u_3} E] →
            {A : Type u_4} →
              [inst_3 : AddMonoid A] →
                [inst_4 : CategoryTheory.HasShift D A] →
                  [inst_5 : CategoryTheory.HasShift E A] →
                    (F : CategoryTheory.SingleFunctors C E A) →
                      (G : CategoryTheory.Functor D E) →
                        [G.CommShift A] →
                          [G.Full] →
                            [G.Faithful] →
                              (Φ : A → CategoryTheory.Functor C D) →
                                ((a : A) → (Φ a).comp G ≅ F.functor a) → CategoryTheory.SingleFunctors C D A

Let C, D and E be categories. Let A be an additive monoid. Assume that D and E have shifts by A and that we have a fully faithful functor G : D ⥤ A which commutes with shifts. Given F : SingleFunctors C E A, and a family of functors Φ a : C ⥤ D with isomorphisms Φ a ⋙ G ≅ F.functor a for all a : A, this is a term in SingleFunctors C D A which is given by the functors Φ a for all a.

Defined in

Mathlib.CategoryTheory.Shift.SingleFunctorsLift

Cited by

5 results in Mathlib

Foundations

Depth 38 from the axioms · uses propext, Classical.choice, Quot.sound

Assumes

CategoryTheory.CategoryCategoryTheory.CategoryCategoryTheory.CategoryAddMonoidCategoryTheory.HasShiftCategoryTheory.HasShiftCategoryTheory.Functor.CommShiftCategoryTheory.Functor.FullCategoryTheory.Functor.Faithful