Nilpotent Category of Abelian Category and Self-Adjoint Functors

Let $\mathcal{C}$ be an additive category. The nilpotent category $\mathrm{Nil} (\mathcal{C})$ of $\mathcal{C}$, consists of objects pairs $(X, x)$ with $X\in\mathcal{C}, x\in\mathrm{End}_{\mathcal{C}}(X)$ such that $x^n=0$ for some positive integer $n$, and a morphism $f:(X, x)\rightarrow (Y,y)$ is $f\in \mathrm{Hom}_{\mathcal{C}}(X, Y)$ satisfying $fx=yf$. A general theory of $\mathrm{Nil}(\mathcal{C})$ is established and it is abelian in the case that $\mathcal{C}$ is abelian. Two abelian categories are equivalent if and only if their nilpotent categories are equivalent, which generalizes a Song, Wu, and Zhang's result. As an application, it is proved all self-adjoint functors are naturally isomorphic to $\mathrm{Hom}$ and $\mathrm{Tensor}$ functors over the category $\mathrm{Nil}$ of finite-dimensional vector spaces. Both $\mathrm{Hom}$ and $\mathrm{Tensor}$ can be naturally generalized to $\mathrm{HOM}$ and $\mathrm{Tensor}$ functor over $\mathrm{Nil}(\mathcal{V})$. They are still self-adjoint, but intrinsically different.