Transport rigid structures over a monoidal equivalence.

def CategoryTheory.exactPairingOfFaithful {C : Type u_1} {D : Type u_2} [CategoryTheory.Category.{u_3, u_1} C] [CategoryTheory.Category.{u_4, u_2} D] [CategoryTheory.MonoidalCategory C] [CategoryTheory.MonoidalCategory D] (F : CategoryTheory.Functor C D) [F.Monoidal] [F.Faithful] {X Y : C} (eval : CategoryTheory.MonoidalCategory.tensorObj Y X ⟶ 𝟙_ C) (coeval : 𝟙_ C ⟶ CategoryTheory.MonoidalCategory.tensorObj X Y) [CategoryTheory.ExactPairing (F.obj X) (F.obj Y)] (map_eval : F.map eval = CategoryTheory.CategoryStruct.comp (CategoryTheory.Functor.OplaxMonoidal.δ F Y X) (CategoryTheory.CategoryStruct.comp (ε_ (F.obj X) (F.obj Y)) (CategoryTheory.Functor.LaxMonoidal.ε F))) (map_coeval : F.map coeval = CategoryTheory.CategoryStruct.comp (CategoryTheory.Functor.OplaxMonoidal.η F) (CategoryTheory.CategoryStruct.comp (η_ (F.obj X) (F.obj Y)) (CategoryTheory.Functor.LaxMonoidal.μ F X Y))) : CategoryTheory.ExactPairing X Y

Given candidate data for an exact pairing, which is sent by a faithful monoidal functor to an exact pairing, the equations holds automatically.

Equations

• CategoryTheory.exactPairingOfFaithful F eval coeval map_eval map_coeval = { coevaluation' := coeval, evaluation' := eval, coevaluation_evaluation' := ⋯, evaluation_coevaluation' := ⋯ }

def CategoryTheory.exactPairingOfFullyFaithful {C : Type u_1} {D : Type u_2} [CategoryTheory.Category.{u_3, u_1} C] [CategoryTheory.Category.{u_4, u_2} D] [CategoryTheory.MonoidalCategory C] [CategoryTheory.MonoidalCategory D] (F : CategoryTheory.Functor C D) [F.Monoidal] [F.Full] [F.Faithful] (X Y : C) [CategoryTheory.ExactPairing (F.obj X) (F.obj Y)] : CategoryTheory.ExactPairing X Y

Given a pair of objects which are sent by a fully faithful functor to a pair of objects with an exact pairing, we get an exact pairing.

Equations

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

def CategoryTheory.hasLeftDualOfEquivalence {C : Type u_1} {D : Type u_2} [CategoryTheory.Category.{u_3, u_1} C] [CategoryTheory.Category.{u_4, u_2} D] [CategoryTheory.MonoidalCategory C] [CategoryTheory.MonoidalCategory D] {F : CategoryTheory.Functor C D} [F.Monoidal] {G : CategoryTheory.Functor D C} (adj : F ⊣ G) [F.IsEquivalence] (X : C) [CategoryTheory.HasLeftDual (F.obj X)] : CategoryTheory.HasLeftDual X

Pull back a left dual along an equivalence.

Equations

• CategoryTheory.hasLeftDualOfEquivalence adj X = CategoryTheory.HasLeftDual.mk (G.obj ᘁ(F.obj X))

def CategoryTheory.hasRightDualOfEquivalence {C : Type u_1} {D : Type u_2} [CategoryTheory.Category.{u_3, u_1} C] [CategoryTheory.Category.{u_4, u_2} D] [CategoryTheory.MonoidalCategory C] [CategoryTheory.MonoidalCategory D] {F : CategoryTheory.Functor C D} [F.Monoidal] {G : CategoryTheory.Functor D C} (adj : F ⊣ G) [F.IsEquivalence] (X : C) [CategoryTheory.HasRightDual (F.obj X)] : CategoryTheory.HasRightDual X

Pull back a right dual along an equivalence.

Equations

• CategoryTheory.hasRightDualOfEquivalence adj X = CategoryTheory.HasRightDual.mk (G.obj (F.obj X)ᘁ)

def CategoryTheory.leftRigidCategoryOfEquivalence {C : Type u_1} {D : Type u_2} [CategoryTheory.Category.{u_3, u_1} C] [CategoryTheory.Category.{u_4, u_2} D] [CategoryTheory.MonoidalCategory C] [CategoryTheory.MonoidalCategory D] {F : CategoryTheory.Functor C D} [F.Monoidal] {G : CategoryTheory.Functor D C} (adj : F ⊣ G) [F.IsEquivalence] [CategoryTheory.LeftRigidCategory D] : CategoryTheory.LeftRigidCategory C

Pull back a left rigid structure along an equivalence.

Equations

• CategoryTheory.leftRigidCategoryOfEquivalence adj = CategoryTheory.LeftRigidCategory.mk

def CategoryTheory.rightRigidCategoryOfEquivalence {C : Type u_1} {D : Type u_2} [CategoryTheory.Category.{u_3, u_1} C] [CategoryTheory.Category.{u_4, u_2} D] [CategoryTheory.MonoidalCategory C] [CategoryTheory.MonoidalCategory D] {F : CategoryTheory.Functor C D} [F.Monoidal] {G : CategoryTheory.Functor D C} (adj : F ⊣ G) [F.IsEquivalence] [CategoryTheory.RightRigidCategory D] : CategoryTheory.RightRigidCategory C

Pull back a right rigid structure along an equivalence.

Equations

• CategoryTheory.rightRigidCategoryOfEquivalence adj = CategoryTheory.RightRigidCategory.mk

def CategoryTheory.rigidCategoryOfEquivalence {C : Type u_1} {D : Type u_2} [CategoryTheory.Category.{u_3, u_1} C] [CategoryTheory.Category.{u_4, u_2} D] [CategoryTheory.MonoidalCategory C] [CategoryTheory.MonoidalCategory D] {F : CategoryTheory.Functor C D} [F.Monoidal] {G : CategoryTheory.Functor D C} (adj : F ⊣ G) [F.IsEquivalence] [CategoryTheory.RigidCategory D] : CategoryTheory.RigidCategory C

Pull back a rigid structure along an equivalence.

Equations

• CategoryTheory.rigidCategoryOfEquivalence adj = CategoryTheory.RigidCategory.mk