Nilpotent category of monoidal category and tensor-hom adjunction (Q6185736)

Let \(\mathcal{C}\) be an abelian monoidal category with tensor product \(\otimes\). This paper constructs a well-defined tensor product \(\overset {\mathcal{N}}{\otimes}\) on the nilpotent category \(\mathrm{Nil}(\mathcal{C})\) comparible with \(\otimes\), establishing that there is almost a natural monoidal structure. As an application, the authors prove that the desired tensor-hom adjunction obtains on \(\mathrm{Nil}(\mathcal{C})\) when \(\mathcal{C}\) is the abelian category of finite-dimensional vector spaces over a fixed field.