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arXiv · 2504.17956

Categorical generalization of spectral decomposition

Koki Nishizawa·Yusuke Ide·Norihiro Tsumagari
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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.

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Koki Nishizawa, Yusuke Ide, Norihiro Tsumagari. 2025-04-24. Categorical generalization of spectral decomposition. https://arxiv.org/abs/2504.17956

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