Categorical Realizations of Quivers

TL;DR

This paper introduces a broad framework for categorical realizations of quivers, generalizing existing concepts and establishing fundamental theorems like Krull-Schmidt and Fitting's Lemma within this context.

Contribution

It develops a unified categorical framework for quiver representations, extending classical theorems to this general setting and demonstrating broader applicability.

Findings

Proves a Krull-Schmidt Theorem for categorical realizations of quivers.

Establishes cancellation properties under milder conditions.

Provides a version of Fitting's Lemma for natural transformations.

Abstract

We introduce and study categorical realizations of quivers. This construction generalizes comma categories and includes representations of quivers on categories, twisted representations of quivers and bilinear pairings as special cases. We prove a Krull-Schmidt Theorem in this general context, which results in a Krull-Schmidt Theorem for the special cases just mentioned. We also show that cancellation holds under milder assumptions. Using similar ideas we prove a version of Fitting's Lemma for natural transformations between functors.