Categorical generalization of spectral decomposition
Koki Nishizawa, Yusuke Ide, Norihiro Tsumagari
TL;DR
The paper addresses the problem of extending spectral decomposition from matrices to arrows in semiadditive $\mathbf{CMon}$-categories, establishing multiple equivalent characterizations for these categories and introducing a generalized notion of spectral decomposition for arrows. It develops a framework where arrows decompose analogously to eigen-spectral data via eigenvalues $\lambda_i$ and eigeninjections $\kappa_i$ on a biproduct $c\cong x_1\oplus x_2$, with stability under composition and addition. A central contribution is proving several equivalent formulations of semiadditive $\mathbf{CMon}$-categories and $\mathbf{CMon}$-functors, plus demonstrating that functors preserving the structure also preserve spectral decompositions, with diverse examples including $\mathbf{Mat}(\mathbb{R})$, $\mathbf{Rel}$, and graph-theoretic decompositions like equitable partitions. This provides a unifying, cross-field approach to spectral-type decompositions in categorical settings, enabling cross-domain analysis of graph separations and relational structures. Future work aims to construct explicit projections, injections, and spectral decompositions for given homogeneous arrows within these semiadditive frameworks.
Abstract
In this paper, we give several equivalent characterizations for a category with finite biproducts and the sum operation of arrows, and called categories satisfying these semiadditive $\mathbf{C}\mathbf{Mon}$-categories. This allow us to give equivalent structures without directly confirming the existence of biproducts. Moreover, we define a generalized notion of the spectral decomposition in semiadditive $\mathbf{C}\mathbf{Mon}$-categories. We also define the notion of a semiadditive $\mathbf{C}\mathbf{Mon}$-functor that preserves the spectral decomposition of arrows. Semiadditive $\mathbf{C}\mathbf{Mon}$-categories and semiadditive $\mathbf{C}\mathbf{Mon}$-functors include many examples.