Proof of a Conjecture of a Pitts

Introduction In this paper we prove a conjecture of Andrew Pitts which states that the Beck Chevalley condition holds for lax pullbacks or comma squares of coherent toposes see Theorem below Pitts conjecture was put forward as a way towards the lax descent theorem for coherent toposes Theorem below The latter entails a dual version for pretoposes which was eventually established by Zawadowski in the setting of Makkai s elaborate theory of Stone duality Our results therefore furnish a proof of the lax descent theorem for pretoposes along the lines originally conceived by Pitts As explained in Zawadowski s paper this theorem can be interpreted as a very general de nability result for coherent logic Perhaps surprisingly our proof of Pitts conjecture needs only simple properties of inverse limits and localization of coherent toposes which are all at least implicitly contained in We have tried to give an accesssible presentation of these properties in the rst sections of this paper Moreover our arguments are completely constructive and valid over an arbitrary base topos We would like to point out that independently yet another proof has recently been given of the descent theorem for pretoposes by David Ballard and Bill Boshuck This elegant proof also uses methods of model theory and seems unrelated to our approach The results of this paper were rst announced at the meeting Geometrical and Logical Aspects of Descent Theory at Oberwolfach September We would like to acknowledge the generous support of the Dutch NWO which made possible a visitor s appointment of the second author at Utrecht