Nilpotent Category of Abelian Category and Self-Adjoint Functors

TL;DR

This paper develops a theory of nilpotent categories within additive and abelian categories, showing their properties, equivalences, and applications to self-adjoint functors, including generalizations of Hom and Tensor functors.

Contribution

It introduces and analyzes the nilpotent category of an additive category, proving its abelian property and characterizing category equivalences via nilpotent categories, with applications to self-adjoint functors.

Findings

Nilpotent categories are abelian if the base category is abelian.

Category equivalences are characterized by their nilpotent categories.

Self-adjoint functors over Nil are isomorphic to Hom and Tensor, which can be generalized.

Abstract

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.