Proper Functors and Fixed Points for Finite Behaviour

Abstract

The rational fixed point of a set functor is well-known to capture thebehaviour of finite coalgebras. In this paper we consider functors on algebraiccategories. 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'snotion of a proper semiring, we introduce the notion of a proper functor. Weshow that for proper functors the rational fixed point is determined as thecolimit of all coalgebras with a free finitely generated algebra as carrier andit is a subcoalgebra of the final coalgebra. Moreover, we prove that a functoris proper if and only if that colimit is a subcoalgebra of the final coalgebra.These results serve as technical tools for soundness and completeness proofsfor coalgebraic regular expression calculi, e.g. for weighted automata