Theorems · Definition · category theory

CategoryTheory.Localization.SmallShiftedHom.equiv

{C : Type u₁} →
  [inst : CategoryTheory.Category.{v₁, u₁} C] →
    {D : Type u₂} →
      [inst_1 : CategoryTheory.Category.{v₂, u₂} D] →
        (W : CategoryTheory.MorphismProperty C) →
          {M : Type w'} →
            [inst_2 : AddMonoid M] →
              [inst_3 : CategoryTheory.HasShift C M] →
                [inst_4 : CategoryTheory.HasShift D M] →
                  (L : CategoryTheory.Functor C D) →
                    [L.IsLocalization W] →
                      [L.CommShift M] →
                        {X Y : C} →
                          [inst_7 : CategoryTheory.Localization.HasSmallLocalizedShiftedHom W M X Y] →
                            {m : M} →
                              CategoryTheory.Localization.SmallShiftedHom W X Y m ≃
                                CategoryTheory.ShiftedHom (L.obj X) (L.obj Y) m

The bijection SmallShiftedHom.{w} W X Y m ≃ ShiftedHom (L.obj X) (L.obj Y) m for all m : M, and X and Y in C when L : C ⥤ D is a localization functor for W : MorphismProperty C such that the category D is equipped with a shift by M and L commutes with the shifts.

Defined in

Mathlib.CategoryTheory.Localization.SmallShiftedHom

Cited by

22 results in Mathlib

Foundations

Depth 92 from the axioms · uses propext, Classical.choice, Quot.sound

Assumes

CategoryTheory.CategoryCategoryTheory.CategoryAddMonoidCategoryTheory.HasShiftCategoryTheory.HasShiftCategoryTheory.Functor.IsLocalizationCategoryTheory.Functor.CommShiftCategoryTheory.Localization.HasSmallLocalizedShiftedHom