On the structure of categories of coalgebras

Consideration of categories of transition systems and related constructions leads to the study of categories of F-coalgebras, where F is an endofuncto r 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 pullback s and generates a cofree comonad, then the category of F-coalgebras is a to pos. Unfortunately, in most of the examples of interest in computer science , the endofunctor F does not preserve pullbacks, though it comes close to d oing so. In this paper we investigate what can be said about the category o f coalgebras under various weakenings of the hypothesis that F preserves pu llbacks. 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 th e category in its effective completion. However, we also show that, in the particular cases of greatest interest, the category of coalgebras is not it self a topos. (C) 2001 Elsevier Science B.V. All rights reserved.