Categorical generalization of spectral decomposition

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.