caa a22 4500 388039132 CHVBK 20180307125016.0 cr unu---uuuuu 161130e199806 xx s 000 0 eng 10.2307/2586836 doi S0022481200014948 pii (NATIONALLICENCE)cambridge-10.2307/2586836 Definability and descent [Elektronische Daten] The present note offers a short argument for the descent theorems of Zawadowski [10] (originally [9]) and Makkai [6], which were conjectured by Pitts after the descent theorem of Joyal and Tierney [3] for open geometric morphisms of (Grothendieck) toposes. The original proofs, which involve variants of Makkai's [5] duality for first order logic, are rather involved and there has been considerable interest in locating simpler proofs. Viewed categorically, the descent theorems establish a bicategorical exactness property (conservative morphisms are effective descent) for pretop (the 2-category of small pretoposes, pretopos functors, and natural transformations), for exact (exact categories, exact functors, and natural transformations), and for bpretop* (Boolean pretoposes, pretopos functors, and natural isomorphisms). Viewed logically, they fragment into a familiar Beth/Tarski-type definability theorem and a covering theorem for certain functors on PCΔ-categories (groupoids, in the Boolean case); the latter (as the former) is a arithmetical statement about the syntax of first order logic ([6, §3])). The argument here involves special models and, independently, a continuity lemma of Makkai. The use of special models is axiomatic in that only a few properties (listed below) are needed. The continuity lemma, 9.1 of [6], is established via forcing and can be read independently of the rest of that paper. Because of its interest to both the model theorist and the category theorist, the argument is first given as straight model theory and afterwards it is briefly indicated how the descent theorems follow. Copyright © Association for Symbolic Logic 1998 Ballard David Department of Mathematics, Sonoma State University, Rohnert Park, CA, USA, E-mail: david.ballard@sonoma.edu Boshuck William Department of Mathematics and Statistics, McGill University, Montreal, PQ, Canada, E-mail: boshuck@math.mcgill.ca The Journal of Symbolic Logic Cambridge University Press 63/2(1998-06), 372-378 0022-4812 63:2<372 1998 63 JSL https://doi.org/10.2307/2586836 text/html Onlinezugriff via DOI 1 research-article jats NATIONALLICENCE 856 40 https://doi.org/10.2307/2586836 text/html Onlinezugriff via DOI NATIONALLICENCE 700 1- Ballard David Department of Mathematics, Sonoma State University, Rohnert Park, CA, USA, E-mail: david.ballard@sonoma.edu NATIONALLICENCE 700 1- Boshuck William Department of Mathematics and Statistics, McGill University, Montreal, PQ, Canada, E-mail: boshuck@math.mcgill.ca NATIONALLICENCE 773 0- The Journal of Symbolic Logic Cambridge University Press 63/2(1998-06), 372-378 0022-4812 63:2<372 1998 63 JSL CC0 http://creativecommons.org/publicdomain/zero/1.0 nationallicence BK010053 XK010053 XK010000 NATIONALLICENCE NATIONALLICENCE NL-cambridge