Hereditary triangular matrix comonads

TL;DR

This paper studies hereditary triangular matrix comonads using a matrix representation approach, simplifying proofs of existing results and generalizing some conditions in the context of additive categories and modules over comonads.

Contribution

It introduces a matrix-based method to analyze comonads, providing simpler proofs and broader conditions for hereditary properties in additive categorical settings.

Findings

Characterization of hereditary triangular matrix comonads

Simplified proofs of Harada's results

Generalization of conditions for homological dimension ≤ 1

Abstract

We recognise Harada's generalized categories of diagrams as a particular case of modules over a monad defined on a finite direct product of additive categories. We work in the dual (albeit formally equivalent) situation, that is, with comodules over comonads. With this conceptual tool at hand, we obtain several of the Harada results with simpler proofs, some of them under more general hypothesis, besides with a characterization of the normal triangular matrix comonads that are hereditary, that is, of homological dimension less or equal than $1$. Our methods rest on a matrix representation of additive functors and natural transformations, which allows us to adapt typical algebraic manipulations from Linear Algebra to the additive categorical setting.