On the structure of categories of coalgebras

Abstract Consideration of categories of transition systems and related constructions leads to the study of categories of F -coalgebras, where F is an endofunctor of the category of sets, or of some more general ‘set-like’ category. It is fairly well known that if E is a topos and F : E → E preserves pullbacks and generates a cofree comonad, then the category of F -coalgebras is a topos. Unfortunately, in most of the examples of interest in computer science, the endofunctor F does not preserve pullbacks, though it comes close to doing so. In this paper we investigate what can be said about the category of coalgebras under various weakenings of the hypothesis that F preserves pullbacks. It turns out that almost all the elementary properties of a topos, except for effectiveness of equivalence relations, are still inherited by the category of coalgebras; and the latter can be recovered by embedding the category in its effective completion. However, we also show that, in the particular cases of greatest interest, the category of coalgebras is not itself a topos.