Since the publication of the paper Carboni and Johnstone (1995), we have become aware of two independent errors in it. Although neither of them has any effect on the main results of the paper, concerning when a category obtained by Artin glueing is a topos, we feel it is appropriate to publish this correction in the hope that it may prevent future readers of Carboni and Johnstone (1995) from being misled. We are grateful to Tom Leinster and to Marek Zawadowski for drawing our attention to the two errors.

Partial equivalence relations (and categories of these) are a standard tool in semantics of type theories and programming languages, since they often provide a cartesian closed category with extended definability. Using the theory of exact categories, we give a category-theoretic explanation of why the construction of a category of partial equivalence relations often produces a cartesian closed category. We show how several familiar examples of categories of partial equivalence relations fit into the general framework.

We consider the following two properties of a functor F from a presheaf topos to the category of sets: (a) F preserves connected limits, and (b) the Artin glueing of F is again a presheaf topos. We show that these two properties are in fact equivalent. In the process, we develop a general technique for associating categorical properties of a category obtained by Artin glueing with preservation properties of the functor along which the glueing takes place. We also give a syntactic characterization of those monads on Set whose functor parts have the above properties, and whose units and multiplications are cartesian natural transformations.