Copower functors

We give a common generalization of two earlier constructions in [H.P. Gumm, T. Schröder, Monoid-labeled transition systems, Electronic Notes in Theoretical Computer Science 44 (1) (2001) 184–203], that yielded coalgebraic type functors for weighted, resp. fuzzy transition systems. Transition labels for these systems were drawn from a commutative monoid M or a complete semilattice L, with the transition structure interacting with the algebraic structure on the labels. Here, we show that those earlier signature functors are in fact instances of a more general construction, provided by the so-called copower functor.Exemplary, we instantiate this functor in categories given by varieties V of algebras. In particular, for the variety S of all semigroups, or the variety M of all (not necessarily commutative) monoids, and with M any monoid, we find that the resulting copower functors MS[−] (resp MM[−]) weakly preserve pullbacks if and only if M is equidivisible (resp. conical and equidivisible).Finally, we show that copower functors are universal in the sense that every faithful Set-functor can be seen as an instance of an appropriate copower functor.