Thanks for the nice counterexample ! I think what you meant is that the constru- ction works for all Heyting algebras (I mean the finitary version : inf-semi-lat tice with exponentiation, no infinite sups or infs are claimed) without a least element. So the archetypical example is the natural numbers with an element for infinity together with the inverse ordering. By the way the display maps in a ll these examples are definable in the sense that whenever f : X -> I is an arbitrary map then there is a greatest subob- ject m : J >--> I such that m*f is a display map. Thus we have definability in the sense of Benabou. In all these cases either m is identity or J = 0 . May it be correct that there are only posetal counterexamples because in this case equalizers do not cause any problems ?? Thomas Streicher ======================================