Adjoint functors on the representation category of $\mathscr{OI}$

TL;DR

This paper explores adjunctions between functors in the representation category of finite linearly ordered sets and injections, and shows the Nakayama functor induces a significant categorical equivalence.

Contribution

It introduces new adjunction relations and proves an equivalence induced by the Nakayama functor in this specific categorical context.

Findings

Established adjunction relations between natural functors.

Proved Nakayama functor induces an equivalence of categories.

Characterized the Serre quotient related to finitely generated modules.

Abstract

In this paper we study adjunction relations between some natural functors on the representation category of the category of finite linearly ordered sets and order-preserving injections. We also prove that the Nakayama functor induces an equivalence from the Serre quotient of the category of finitely generated modules by the category of finitely generated torsion modules to the category of finite dimensional modules.