15 research outputs found
Proper Functors and Fixed Points for Finite Behaviour
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
On Well-Founded and Recursive Coalgebras
This paper studies fundamental questions concerning category-theoretic models
of induction and recursion. We are concerned with the relationship between
well-founded and recursive coalgebras for an endofunctor. For monomorphism
preserving endofunctors on complete and well-powered categories every coalgebra
has a well-founded part, and we provide a new, shorter proof that this is the
coreflection in the category of all well-founded coalgebras. We present a new
more general proof of Taylor's General Recursion Theorem that every
well-founded coalgebra is recursive, and we study under which hypothesis the
converse holds. In addition, we present a new equivalent characterization of
well-foundedness: a coalgebra is well-founded iff it admits a
coalgebra-to-algebra morphism to the initial algebra