Hereditary triangular matrix comonads

Abstract

We recognize 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 than or equal to 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.

Keywords:

• matrix comonads
• categories of comodules
• abelian hereditary categories
• global homological dimension

AMS Subject Classifications:

• 15A30
• 18G20
• 18C20
• 16T15

Notes

No potential conflict of interest was reported by the authors.

Additional information

Funding

This research was supported by the Spanish Ministerio de Economía y Competitividad and the European Union [grant number MTM2013–41992-P].