Nilpotent Category of Monoidal Category and Tensor–Hom Adjunction

Abstract

Let \(\mathcal {C}\) be an abelian monoidal category. It is proved that the nilpotent category \({\text {Nil}}(\mathcal {C})\) of \(\mathcal {C}\) admits almost monoidal structure except the unit axiom. As an application, it is proved that Hom and Tensor functors exist over \({\text {Nil}}(\mathcal {C})\) and tensor–hom adjunction remains true over the nilpotent category of the category of finite-dimensional vector spaces, which develops some recent results on this topic.