The (naive) shift on the opposite category

If C is a category equipped with a shift by a monoid A, the opposite category can be equipped with a shift such that the shift functor by n is (shiftFunctor C n).op. This is the "naive" opposite shift, which we shall set on a category OppositeShift C A, which is a type synonym for Cᵒᵖ.

However, for the application to (pre)triangulated categories, we would like to define the shift on Cᵒᵖ so that shiftFunctor Cᵒᵖ n for n : ℤ identifies to (shiftFunctor C (-n)).op rather than (shiftFunctor C n).op. Then, the construction of the shift on Cᵒᵖ shall combine the shift on OppositeShift C A and another construction of the "pullback" of a shift by a monoid morphism like n ↦ -n.

If F : C ⥤ D is a functor between categories equipped with shifts by A, we define a type synonym OppositeShift.functor A F for F.op. When F has a CommShift structure by A, we define a CommShift structure by A on OppositeShift.functor A F. In this way, we can make this an instance and reserve F.op for the CommShift instance by the modified shift in the case of (pre)triangulated categories.

Similarly,if τ is a natural transformation between functors F,G : C ⥤ D, we define a type synonym for τ.op called OppositeShift.natTrans A τ : OppositeShift.functor A F ⟶ OppositeShift.functor A G. When τ has a CommShift structure by A (i.e. is compatible with CommShift structures on F and G), we define a CommShift structure by A on OppositeShift.natTrans A τ.

Finally, if we have an adjunction F ⊣ G (with G : D ⥤ C), we define a type synonym OppositeShift.adjunction A adj : OppositeShift.functor A G ⊣ OppositeShift.functor A F for adj.op, and we show that, if adj compatible with CommShift structures on F and G, then OppositeShift.adjunction A adj is also compatible with the pulled back CommShift structures.

Given a CommShift structure on a functor F, we define a CommShift structure on F.op (and vice versa). We also prove that, if an adjunction F ⊣ G is compatible with CommShift structures on F and G, then the opposite adjunction G.op ⊣ F.op is compatible with the opposite CommShift structures.

noncomputable def CategoryTheory.HasShift.mkShiftCoreOp (C : Type u_1) [CategoryTheory.Category.{u_3, u_1} C] (A : Type u_2) [AddMonoid A] [CategoryTheory.HasShift C A] : CategoryTheory.ShiftMkCore Cᵒᵖ A

Construction of the naive shift on the opposite category of a category C: the shiftfunctor by n is (shiftFunctor C n).op.

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def CategoryTheory.OppositeShift (C : Type u_1) [CategoryTheory.Category.{u_4, u_1} C] (A : Type u_3) [AddMonoid A] [CategoryTheory.HasShift C A] : Type u_1

The category OppositeShift C A is the opposite category Cᵒᵖ equipped with the naive shift: shiftFunctor (OppositeShift C A) n is (shiftFunctor C n).op.

Equations

• CategoryTheory.OppositeShift C A = Cᵒᵖ

instance CategoryTheory.instCategoryOppositeShift (C : Type u_1) [CategoryTheory.Category.{u_3, u_1} C] (A : Type u_2) [AddMonoid A] [CategoryTheory.HasShift C A] : CategoryTheory.Category.{u_3, u_1} (CategoryTheory.OppositeShift C A)

Equations

• CategoryTheory.instCategoryOppositeShift C A = id inferInstance

noncomputable instance CategoryTheory.instHasShiftOppositeShift (C : Type u_1) [CategoryTheory.Category.{u_3, u_1} C] (A : Type u_2) [AddMonoid A] [CategoryTheory.HasShift C A] : CategoryTheory.HasShift (CategoryTheory.OppositeShift C A) A

Equations

• CategoryTheory.instHasShiftOppositeShift C A = CategoryTheory.hasShiftMk Cᵒᵖ A (CategoryTheory.HasShift.mkShiftCoreOp C A)

instance CategoryTheory.instHasZeroObjectOppositeShift (C : Type u_1) [CategoryTheory.Category.{u_3, u_1} C] (A : Type u_2) [AddMonoid A] [CategoryTheory.HasShift C A] [CategoryTheory.Limits.HasZeroObject C] : CategoryTheory.Limits.HasZeroObject (CategoryTheory.OppositeShift C A)

instance CategoryTheory.instPreadditiveOppositeShift (C : Type u_1) [CategoryTheory.Category.{u_3, u_1} C] (A : Type u_2) [AddMonoid A] [CategoryTheory.HasShift C A] [CategoryTheory.Preadditive C] : CategoryTheory.Preadditive (CategoryTheory.OppositeShift C A)

Equations

• CategoryTheory.instPreadditiveOppositeShift C A = id inferInstance

instance CategoryTheory.instAdditiveOppositeShiftShiftFunctor (C : Type u_1) [CategoryTheory.Category.{u_3, u_1} C] (A : Type u_2) [AddMonoid A] [CategoryTheory.HasShift C A] [CategoryTheory.Preadditive C] (n : A) [(CategoryTheory.shiftFunctor C n).Additive] : (CategoryTheory.shiftFunctor (CategoryTheory.OppositeShift C A) n).Additive

theorem CategoryTheory.oppositeShiftFunctorZero_inv_app (C : Type u_1) [CategoryTheory.Category.{u_3, u_1} C] (A : Type u_2) [AddMonoid A] [CategoryTheory.HasShift C A] (X : CategoryTheory.OppositeShift C A) : (CategoryTheory.shiftFunctorZero (CategoryTheory.OppositeShift C A) A).inv.app X = ((CategoryTheory.shiftFunctorZero C A).hom.app (Opposite.unop X)).op

theorem CategoryTheory.oppositeShiftFunctorZero_hom_app (C : Type u_1) [CategoryTheory.Category.{u_3, u_1} C] (A : Type u_2) [AddMonoid A] [CategoryTheory.HasShift C A] (X : CategoryTheory.OppositeShift C A) : (CategoryTheory.shiftFunctorZero (CategoryTheory.OppositeShift C A) A).hom.app X = ((CategoryTheory.shiftFunctorZero C A).inv.app (Opposite.unop X)).op

theorem CategoryTheory.oppositeShiftFunctorAdd_inv_app {C : Type u_1} [CategoryTheory.Category.{u_3, u_1} C] {A : Type u_2} [AddMonoid A] [CategoryTheory.HasShift C A] (X : CategoryTheory.OppositeShift C A) (a b : A) : (CategoryTheory.shiftFunctorAdd (CategoryTheory.OppositeShift C A) a b).inv.app X = ((CategoryTheory.shiftFunctorAdd C a b).hom.app (Opposite.unop X)).op

theorem CategoryTheory.oppositeShiftFunctorAdd_hom_app {C : Type u_1} [CategoryTheory.Category.{u_3, u_1} C] {A : Type u_2} [AddMonoid A] [CategoryTheory.HasShift C A] (X : CategoryTheory.OppositeShift C A) (a b : A) : (CategoryTheory.shiftFunctorAdd (CategoryTheory.OppositeShift C A) a b).hom.app X = ((CategoryTheory.shiftFunctorAdd C a b).inv.app (Opposite.unop X)).op

theorem CategoryTheory.oppositeShiftFunctorAdd'_inv_app {C : Type u_1} [CategoryTheory.Category.{u_3, u_1} C] {A : Type u_2} [AddMonoid A] [CategoryTheory.HasShift C A] (X : CategoryTheory.OppositeShift C A) (a b c : A) (h : a + b = c) : (CategoryTheory.shiftFunctorAdd' (CategoryTheory.OppositeShift C A) a b c h).inv.app X = ((CategoryTheory.shiftFunctorAdd' C a b c h).hom.app (Opposite.unop X)).op

theorem CategoryTheory.oppositeShiftFunctorAdd'_hom_app {C : Type u_1} [CategoryTheory.Category.{u_3, u_1} C] {A : Type u_2} [AddMonoid A] [CategoryTheory.HasShift C A] (X : CategoryTheory.OppositeShift C A) (a b c : A) (h : a + b = c) : (CategoryTheory.shiftFunctorAdd' (CategoryTheory.OppositeShift C A) a b c h).hom.app X = ((CategoryTheory.shiftFunctorAdd' C a b c h).inv.app (Opposite.unop X)).op

def CategoryTheory.OppositeShift.functor {C : Type u_1} {D : Type u_2} [CategoryTheory.Category.{u_4, u_1} C] [CategoryTheory.Category.{u_5, u_2} D] (A : Type u_3) [AddMonoid A] [CategoryTheory.HasShift C A] [CategoryTheory.HasShift D A] (F : CategoryTheory.Functor C D) : CategoryTheory.Functor (CategoryTheory.OppositeShift C A) (CategoryTheory.OppositeShift D A)

The functor F.op, seen as a functor from OppositeShift C A to OppositeShift D A. (We will use this to carry a CommShift instance for the naive shifts on the opposite category. Then, in the pretriangulated case, we will be able to put a CommShift instance on F.op for the modified shifts and not deal with instance clashes.

Equations

• CategoryTheory.OppositeShift.functor A F = F.op

def CategoryTheory.OppositeShift.natTrans {C : Type u_1} {D : Type u_2} [CategoryTheory.Category.{u_4, u_1} C] [CategoryTheory.Category.{u_5, u_2} D] (A : Type u_3) [AddMonoid A] [CategoryTheory.HasShift C A] [CategoryTheory.HasShift D A] {F G : CategoryTheory.Functor C D} (τ : F ⟶ G) : CategoryTheory.OppositeShift.functor A G ⟶ CategoryTheory.OppositeShift.functor A F

The natural transformation τ, seen as a natural transformation from OppositeShift.functor F A to OppositeShift.functor G A..

Equations

• CategoryTheory.OppositeShift.natTrans A τ = CategoryTheory.NatTrans.op τ

noncomputable instance CategoryTheory.Functor.commShiftOp {C : Type u_1} {D : Type u_2} [CategoryTheory.Category.{u_4, u_1} C] [CategoryTheory.Category.{u_5, u_2} D] (A : Type u_3) [AddMonoid A] [CategoryTheory.HasShift C A] [CategoryTheory.HasShift D A] (F : CategoryTheory.Functor C D) [F.CommShift A] : (CategoryTheory.OppositeShift.functor A F).CommShift A

Given a CommShift structure on F, this is the corresponding CommShift structure on OppositeShift.functor F (for the naive shifts on the opposite categories).

Equations

• CategoryTheory.Functor.commShiftOp A F = { iso := fun (a : A) => (CategoryTheory.NatIso.op (F.commShiftIso a)).symm, zero := ⋯, add := ⋯ }

theorem CategoryTheory.Functor.commShiftOp_iso_eq {C : Type u_1} {D : Type u_2} [CategoryTheory.Category.{u_4, u_1} C] [CategoryTheory.Category.{u_5, u_2} D] (A : Type u_3) [AddMonoid A] [CategoryTheory.HasShift C A] [CategoryTheory.HasShift D A] (F : CategoryTheory.Functor C D) [F.CommShift A] (a : A) : (CategoryTheory.OppositeShift.functor A F).commShiftIso a = (CategoryTheory.NatIso.op (F.commShiftIso a)).symm

noncomputable def CategoryTheory.Functor.commShiftUnop {C : Type u_1} {D : Type u_2} [CategoryTheory.Category.{u_4, u_1} C] [CategoryTheory.Category.{u_5, u_2} D] (A : Type u_3) [AddMonoid A] [CategoryTheory.HasShift C A] [CategoryTheory.HasShift D A] (F : CategoryTheory.Functor C D) [(CategoryTheory.OppositeShift.functor A F).CommShift A] : F.CommShift A

Given a CommShift structure on OppositeShift.functor F (for the naive shifts on the opposite categories), this is the corresponding CommShift structure on F.

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@[simp]

theorem CategoryTheory.Functor.commShiftUnop_iso {C : Type u_1} {D : Type u_2} [CategoryTheory.Category.{u_4, u_1} C] [CategoryTheory.Category.{u_5, u_2} D] (A : Type u_3) [AddMonoid A] [CategoryTheory.HasShift C A] [CategoryTheory.HasShift D A] (F : CategoryTheory.Functor C D) [(CategoryTheory.OppositeShift.functor A F).CommShift A] (a : A) : CategoryTheory.Functor.CommShift.iso a = CategoryTheory.NatIso.removeOp ((CategoryTheory.OppositeShift.functor A F).commShiftIso a).symm

instance CategoryTheory.NatTrans.commShift_op {C : Type u_1} {D : Type u_2} [CategoryTheory.Category.{u_5, u_1} C] [CategoryTheory.Category.{u_4, u_2} D] (A : Type u_3) [AddMonoid A] [CategoryTheory.HasShift C A] [CategoryTheory.HasShift D A] {F G : CategoryTheory.Functor C D} [F.CommShift A] [G.CommShift A] (τ : F ⟶ G) [CategoryTheory.NatTrans.CommShift τ A] : CategoryTheory.NatTrans.CommShift (CategoryTheory.OppositeShift.natTrans A τ) A

def CategoryTheory.NatTrans.OppositeShift.natIsoId (C : Type u_1) [CategoryTheory.Category.{u_4, u_1} C] (A : Type u_3) [AddMonoid A] [CategoryTheory.HasShift C A] : CategoryTheory.Functor.id (CategoryTheory.OppositeShift C A) ≅ CategoryTheory.OppositeShift.functor A (CategoryTheory.Functor.id C)

The obvious isomorphism between the identity of OppositeShift C A and OppositeShift.functor (𝟙 C).

Equations

• CategoryTheory.NatTrans.OppositeShift.natIsoId C A = CategoryTheory.Iso.refl (CategoryTheory.Functor.id (CategoryTheory.OppositeShift C A))

instance CategoryTheory.NatTrans.instCommShiftOppositeShiftHomFunctorNatIsoId {C : Type u_1} [CategoryTheory.Category.{u_4, u_1} C] (A : Type u_3) [AddMonoid A] [CategoryTheory.HasShift C A] : CategoryTheory.NatTrans.CommShift (CategoryTheory.NatTrans.OppositeShift.natIsoId C A).hom A

The natural isomorphism NatTrans.OppositeShift.natIsoId C A commutes with shifts.

def CategoryTheory.NatTrans.OppositeShift.natIsoComp {C : Type u_1} {D : Type u_2} [CategoryTheory.Category.{u_5, u_1} C] [CategoryTheory.Category.{u_6, u_2} D] (A : Type u_3) [AddMonoid A] [CategoryTheory.HasShift C A] [CategoryTheory.HasShift D A] (F : CategoryTheory.Functor C D) {E : Type u_4} [CategoryTheory.Category.{u_7, u_4} E] [CategoryTheory.HasShift E A] (G : CategoryTheory.Functor D E) : CategoryTheory.OppositeShift.functor A (F.comp G) ≅ (CategoryTheory.OppositeShift.functor A F).comp (CategoryTheory.OppositeShift.functor A G)

The obvious isomorphism between OppositeShift.functor (F ⋙ G) and the composition of OppositeShift.functor F and OppositeShift.functor G.

Equations

• CategoryTheory.NatTrans.OppositeShift.natIsoComp A F G = CategoryTheory.Iso.refl (CategoryTheory.OppositeShift.functor A (F.comp G))

instance CategoryTheory.NatTrans.instCommShiftOppositeShiftHomFunctorNatIsoComp {C : Type u_1} {D : Type u_2} [CategoryTheory.Category.{u_5, u_1} C] [CategoryTheory.Category.{u_6, u_2} D] (A : Type u_3) [AddMonoid A] [CategoryTheory.HasShift C A] [CategoryTheory.HasShift D A] (F : CategoryTheory.Functor C D) {E : Type u_4} [CategoryTheory.Category.{u_7, u_4} E] [CategoryTheory.HasShift E A] (G : CategoryTheory.Functor D E) [F.CommShift A] [G.CommShift A] : CategoryTheory.NatTrans.CommShift (CategoryTheory.NatTrans.OppositeShift.natIsoComp A F G).hom A

def CategoryTheory.OppositeShift.adjunction {C : Type u_1} {D : Type u_2} [CategoryTheory.Category.{u_4, u_1} C] [CategoryTheory.Category.{u_5, u_2} D] (A : Type u_3) [AddMonoid A] [CategoryTheory.HasShift C A] [CategoryTheory.HasShift D A] {F : CategoryTheory.Functor C D} {G : CategoryTheory.Functor D C} (adj : F ⊣ G) : CategoryTheory.OppositeShift.functor A G ⊣ CategoryTheory.OppositeShift.functor A F

The adjunction adj, seen as an adjunction between OppositeShift.functor G and OppositeShift.functor F.

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• One or more equations did not get rendered due to their size.

theorem CategoryTheory.OppositeShift.adjunction_counit {C : Type u_1} {D : Type u_2} [CategoryTheory.Category.{u_4, u_1} C] [CategoryTheory.Category.{u_5, u_2} D] (A : Type u_3) [AddMonoid A] [CategoryTheory.HasShift C A] [CategoryTheory.HasShift D A] {F : CategoryTheory.Functor C D} {G : CategoryTheory.Functor D C} (adj : F ⊣ G) : (CategoryTheory.OppositeShift.adjunction A adj).counit = CategoryTheory.CategoryStruct.comp (CategoryTheory.NatTrans.OppositeShift.natIsoComp A F G).inv (CategoryTheory.CategoryStruct.comp (CategoryTheory.OppositeShift.natTrans A adj.unit) (CategoryTheory.NatTrans.OppositeShift.natIsoId C A).inv)

theorem CategoryTheory.OppositeShift.adjunction_unit {C : Type u_1} {D : Type u_2} [CategoryTheory.Category.{u_4, u_1} C] [CategoryTheory.Category.{u_5, u_2} D] (A : Type u_3) [AddMonoid A] [CategoryTheory.HasShift C A] [CategoryTheory.HasShift D A] {F : CategoryTheory.Functor C D} {G : CategoryTheory.Functor D C} (adj : F ⊣ G) : (CategoryTheory.OppositeShift.adjunction A adj).unit = CategoryTheory.CategoryStruct.comp (CategoryTheory.NatTrans.OppositeShift.natIsoId D A).hom (CategoryTheory.CategoryStruct.comp (CategoryTheory.OppositeShift.natTrans A adj.counit) (CategoryTheory.NatTrans.OppositeShift.natIsoComp A G F).hom)

instance CategoryTheory.Adjunction.commShift_op {C : Type u_1} {D : Type u_2} [CategoryTheory.Category.{u_4, u_1} C] [CategoryTheory.Category.{u_5, u_2} D] (A : Type u_3) [AddMonoid A] [CategoryTheory.HasShift C A] [CategoryTheory.HasShift D A] {F : CategoryTheory.Functor C D} {G : CategoryTheory.Functor D C} (adj : F ⊣ G) [F.CommShift A] [G.CommShift A] [adj.CommShift A] : (CategoryTheory.OppositeShift.adjunction A adj).CommShift A

If an adjunction F ⊣ G is compatible with CommShift structures on F and G, then the opposite adjunction OppositeShift.adjunction adj is compatible with the opposite CommShift structures.