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.MonoidalFunctor C D) [CategoryTheory.Faithful F.toFunctor] {X : C} {Y : C} (eval : CategoryTheory.MonoidalCategory.tensorObj Y X ⟶ CategoryTheory.MonoidalCategory.tensorUnit C) (coeval : CategoryTheory.MonoidalCategory.tensorUnit C ⟶ CategoryTheory.MonoidalCategory.tensorObj X Y) [CategoryTheory.ExactPairing (F.obj X) (F.obj Y)] (map_eval : F.map eval = CategoryTheory.CategoryStruct.comp (CategoryTheory.inv (CategoryTheory.LaxMonoidalFunctor.μ F.toLaxMonoidalFunctor Y X)) (CategoryTheory.CategoryStruct.comp (ε_ (F.obj X) (F.obj Y)) F.ε)) (map_coeval : F.map coeval = CategoryTheory.CategoryStruct.comp (CategoryTheory.inv F.ε) (CategoryTheory.CategoryStruct.comp (η_ (F.obj X) (F.obj Y)) (CategoryTheory.LaxMonoidalFunctor.μ F.toLaxMonoidalFunctor 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.

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.MonoidalFunctor C D) [CategoryTheory.Full F.toFunctor] [CategoryTheory.Faithful F.toFunctor] (X : C) (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.

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.MonoidalFunctor C D) [CategoryTheory.IsEquivalence F.toFunctor] (X : C) [CategoryTheory.HasLeftDual (F.obj X)] : CategoryTheory.HasLeftDual X

Pull back a left dual along an equivalence.

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.MonoidalFunctor C D) [CategoryTheory.IsEquivalence F.toFunctor] (X : C) [CategoryTheory.HasRightDual (F.obj X)] : CategoryTheory.HasRightDual X

Pull back a right dual along an equivalence.

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.MonoidalFunctor C D) [CategoryTheory.IsEquivalence F.toFunctor] [CategoryTheory.LeftRigidCategory D] : CategoryTheory.LeftRigidCategory C

Pull back a left rigid structure along an equivalence.

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.MonoidalFunctor C D) [CategoryTheory.IsEquivalence F.toFunctor] [CategoryTheory.RightRigidCategory D] : CategoryTheory.RightRigidCategory C

Pull back a right rigid structure along an equivalence.

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.MonoidalFunctor C D) [CategoryTheory.IsEquivalence F.toFunctor] [CategoryTheory.RigidCategory D] : CategoryTheory.RigidCategory C

Pull back a rigid structure along an equivalence.