Final coalgebras over R-modules
Abstract
Final coalgebras and the coinduction principle for coalgebras over R-modules, where R is a commutative ring, are essential in determining the denotational behaviour of various types of linear weighted automata. The goal of this paper is to provide lower and upper bounds on the ordinals that are needed to construct final coalgebras over R-modules as the limit of a transfinite sequence of functor iterations. We give an example of a binary intersection preserving finitary functor on the category of abelian groups, which is the category of modules over integers, that fails to converge in ω· 2 steps, thus settling an open problem of Adámek, Milius, and Moss. Then we relate this to the failure of the Artinian property of the integers and show that all finitary functors on R-modules for a Noetherian ring R with Krull dimension n, which preserve countable directed intersections and a generalised notion of monomorphism, converge in ω· (n + 2) steps. We also establish how the connection between Aczel-Mendler bisimulation and behavioural equivalence is mediated by the functor Ext1. These results show that homological algebra tools can play an important role in coalgebra.
Related
- Presenting Point-Glued Self-Similar Objects as Final Coalgebras in Metric Spaces via Completion: A Categorical Hutchinson Theorem
- Cotensor decompositions for comodules over graded vertex operator coalgebras
- Fusion rules and rigidity for weight modules over the simple admissible affine sl(2) and N=2 superconformal vertex operator superalgebras
- simple cartan-free modules of finite rank over the super-virasoro algebras
- Splitting of projective modules over some affine algebras
BibTeX
@misc{flynnconnolly2026final,
title = {{Final coalgebras over R-modules}},
author = {Oisín Flynn-Connolly and Henning Basold and Chase Ford},
howpublished = {arXiv},
year = {2026},
month = nov,
doi = {10.48550/arxiv.2611.03174},
eprint = {2611.03174},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2611.03174},
}
RIS
TY - UNPB
TI - Final coalgebras over R-modules
AU - Oisín Flynn-Connolly
AU - Henning Basold
AU - Chase Ford
PY - 2026
DA - 2026/11/02
PB - arXiv
DO - 10.48550/arxiv.2611.03174
UR - https://arxiv.org/abs/2611.03174
AB - Final coalgebras and the coinduction principle for coalgebras over R-modules, where R is a commutative ring, are essential in determining the denotational behaviour of various types of linear weighted automata. The goal of this paper is to provide lower and upper bounds on the ordinals that are needed to construct final coalgebras over R-modules as the limit of a transfinite sequence of functor iterations. We give an example of a binary intersection preserving finitary functor on the category of abelian groups, which is the category of modules over integers, that fails to converge in ω· 2 steps, thus settling an open problem of Adámek, Milius, and Moss. Then we relate this to the failure of the Artinian property of the integers and show that all finitary functors on R-modules for a Noetherian ring R with Krull dimension n, which preserve countable directed intersections and a generalised notion of monomorphism, converge in ω· (n + 2) steps. We also establish how the connection between Aczel-Mendler bisimulation and behavioural equivalence is mediated by the functor Ext^1. These results show that homological algebra tools can play an important role in coalgebra.
ER -