Spectra of categories
Let \({\mathcal C}\) be a small category. The authors define the Laplacian of \({\mathcal C}\) via \(\Delta({\mathcal C})= (\#\Hom_{\mathcal C} (X,Y))\), \(X,Y\in Ob({\mathcal C})\), where \(Ob({\mathcal C})\) is the set of objects of \({\mathcal C}\), and \(\#\) denotes the cardinality. The following interesting results are obtained: (1) For finite abelian groups (i.e., \(\mathbb{Z}\)-modules) \(A_1,A_2, \dots,A_n\), \(\Delta(A_1, \dots,A_n)= (\#\Hom (A_i,A_j))\) is a symmetric matrix; (2) Let \(\mathbb{F}_q\) be the finite field of \(q\) elements, then \(\Delta_{\mathbb{F}_q} (\mathbb{F}_q^{m_1}, \dots, \mathbb{F}_q^{m_n})= (q^{m_i}q^{m_j})\) is a semi-positive matrix for integers \(m_i\geq 0\); (3) Let \(1\leq k_1<k_2 <\cdots< k_n<\dots\) be coprime integers, and the spectrum of \(\Delta\), i.e., the set of eigenvalues of \(\Delta (\mathbb{Z}/k_1 \mathbb{Z},\dots, \mathbb{Z}/ k_n \mathbb{Z})\), \(\text{Spect} (\mathbb{Z}/k_1 \mathbb{Z},\dots, \mathbb{Z}/k_n)= \{\lambda_1^{(n) }, \dots, \lambda_n^{(n)}\}\). These eigenvalues are all simple and located as \(k_1-1 <\lambda_1^{(n)} <k_2-1<\lambda_2^{(n)}< \cdots<k_n-1< \lambda_n^{(n)}\). Moreover, for each fixed \(m\geq 1\), \(\lim_{n\to \infty} \lambda_m^{(n)}\) exists and \(\lim_{n\to \infty} \lambda_m^{(n)}= k_m-1\), if \(\sum^\infty_{n=1} {1\over k_n} =+\infty\); \(\lim_{n \to \infty} \lambda_m^{(n)} >k_m-1\), if \(\sum^\infty_{n=1} {1\over k_n} <+\infty\).
scientific article; zbMATH DE number 1375159