Homology Groups and Categorical Diagonalization

TL;DR

This paper explores the connection between homology groups and categorical diagonalization, showing how (co)homology can be viewed as eigenvalues within a categorical framework, with implications for chain complex analysis.

Contribution

It introduces a novel perspective linking (co)homology groups to eigenvalues in categorical diagonalization, expanding the understanding of chain complexes in algebraic topology.

Findings

Homology groups correspond to eigenvalues in categorical diagonalization.

Fixed objects are isomorphic to (co)homology groups under certain chain map conditions.

(Co)homology groups can be viewed as eigenvalues of chain complexes.

Abstract

We discuss the relationship between (co)homology groups and categorical diagonalization. We consider the category of chain complexes in the category of finitely generated free modules on a commutative ring. For a fixed chain complex with zero maps as an object, a chain map from the object to another chain complex is defined, and the chain map introduce a mapping cone. We found that the fixed object is isomorphic to the (co)homology groups of the codomain of the chain map if and only if the chain map is injective to the cokernel of differentials of the codomain chain complex and the mapping cone is homotopy equivalent to zero. On the other hand, the fixed object is regarded as a categorified eigenvalue of the chain complex in the context of the categorical diagonalization introduced by B.Elias and M. Hogancamp arXiv:1801.00191v1. It is found that (co)homology groups are constructed as the eigenvalue of a chain complex.