Theorems · Definition · category theory

CategoryTheory.Functor.ShiftSequence.induced

{C : Type u_1} →
  {D : Type u_2} →
    {A : Type u_3} →
      [inst : CategoryTheory.Category.{v_1, u_1} C] →
        [inst_1 : CategoryTheory.Category.{v_2, u_2} D] →
          [inst_2 : CategoryTheory.Category.{v_3, u_3} A] →
            {L : CategoryTheory.Functor C D} →
              {F : CategoryTheory.Functor D A} →
                {G : CategoryTheory.Functor C A} →
                  (L.comp F ≅ G) →
                    (M : Type u_4) →
                      [inst_3 : AddMonoid M] →
                        [inst_4 : CategoryTheory.HasShift C M] →
                          [inst_5 : G.ShiftSequence M] →
                            (F' : M → CategoryTheory.Functor D A) →
                              ((m : M) → L.comp (F' m) ≅ G.shift m) →
                                [((CategoryTheory.Functor.whiskeringLeft C D A).obj L).Full] →
                                  [((CategoryTheory.Functor.whiskeringLeft C D A).obj L).Faithful] →
                                    [inst_8 : CategoryTheory.HasShift D M] → [L.CommShift M] → F.ShiftSequence M

Given an isomorphism of functors e : L ⋙ F ≅ G relating functors L : C ⥤ D, F : D ⥤ A and G : C ⥤ A, an additive monoid M, a family of functors F' : M → D ⥤ A equipped with isomorphisms e' : ∀ m, L ⋙ F' m ≅ G.shift m, this is the shift sequence induced on F induced by a shift sequence for the functor G, provided that the functor (whiskeringLeft C D A).obj L of precomposition by L is fully faithful.

Defined in

Mathlib.CategoryTheory.Shift.InducedShiftSequence

Cited by

6 results in Mathlib

Foundations

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

Assumes

CategoryTheory.CategoryCategoryTheory.CategoryCategoryTheory.CategoryAddMonoidCategoryTheory.HasShiftCategoryTheory.Functor.ShiftSequenceCategoryTheory.Functor.FullCategoryTheory.Functor.FaithfulCategoryTheory.HasShiftCategoryTheory.Functor.CommShift