Proper Functors and Fixed Points for Finite Behaviour
Abstract
The rational fixed point of a set functor is well-known to capture the behaviour of finite coalgebras. In this paper we consider functors on algebraic categories. For them the rational fixed point may no longer be fully abstract, i.e. a subcoalgebra of the final coalgebra. Inspired by \'Esik and Maletti's notion of a proper semiring, we introduce the notion of a proper functor. We show that for proper functors the rational fixed point is determined as the colimit of all coalgebras with a free finitely generated algebra as carrier and it is a subcoalgebra of the final coalgebra. Moreover, we prove that a functor is proper if and only if that colimit is a subcoalgebra of the final coalgebra. These results serve as technical tools for soundness and completeness proofs for coalgebraic regular expression calculi, e.g. for weighted automata.
Keywords
Cite
@article{arxiv.1705.09198,
title = {Proper Functors and Fixed Points for Finite Behaviour},
author = {Stefan Milius},
journal= {arXiv preprint arXiv:1705.09198},
year = {2023}
}