Commutation Structures

For fixed object X in a monoidal category, an X-commutation structure on an object A is just a map X⊗A → A⊗X. We study aspects of such structures in case A has a dual object. We consider a monoidal category V,⊗, I; for simplicity we let it be strict (the application we have in mind is anyway a category of endofuncors on a category, with composition as ⊗). Let X be an object in V, fixed throughout. An X-commutation structure on an object A ∈ V is an arrow α : X ⊗ A → A⊗X. A morphism of X-commutation structures (A, α) → (B, β) is an arrow A → B such that the obvious square X ⊗ A αA⊗X X ⊗ B X ⊗ f ? β B ⊗X f ⊗X ? commutes. In this way, we get a category of X-commutation structures; denote it X-V. There is a faithful forgetful functor X-V → V: to (A, α), associate A. There is also a monoidal structure on X-V, preserved strictly by the forgetful functor X-V → V: If (A, α) and (B, β) are objects in X-V, we get a commutation structure γ on A⊗B in an obvious way: γ is taken to be the composite X ⊗ A⊗ B α⊗ BA⊗X ⊗ B A⊗ β A⊗ B ⊗X.