Proof of a conjecture of A. Pitts
type: Item
instance of: scholarly article
title: Proof of a conjecture of A. Pitts
zbMATH Open document ID: 0944.18004
DOI: 10.1016/S0022-4049(98)00118-2
author: J. J. C. Vermeulen
author: Ieke Moerdijk
published in: Journal of Pure and Applied Algebra
publication date: 2000-09-17T00:00:00Z
review text: The subject of lax descent for several classes of morphisms has been extensively investigated by \textit{A. M. Pitts} in several lectures at conferences in Category Theory but are mostly unpublished. A well-known theorem of this kind [\textit{A. M. Pitts}, ``Lax descent for geometric morphisms'', Lecture, Cambridge Categories Conference, July 1986)] says that if the lower leg in a lax pullback of toposes over a base topos \({\mathcal S}\) is essential, then the geometric morphism opposite it is locally connected and moreover the Beck-Chevalley condition is satisfied. Its original proof is carried out by looking at the theories which the toposes in question classify so that, in this sense, it becomes a (definability) theorem in logic. A different proof of this result [\textit{M. Bunge} and \textit{J. Funk}, ``Spreads and the symmetric topos'', J. Pure Appl. Algebra 113, No. 1, 1-38 (1996; Zbl 0861.18004)] employs instead the topos classifying \({\mathcal S}\)-valued distributions on toposes bounded over \(\mathcal S\) and the explicit construction of the complete spread associated with an \(\mathcal S\)-valued distribution. The corresponding ``lax descent theorem'' for coherent toposes (actually, coherence morphisms in the relative version) was conjectured by Pitts who pointed out that the key result needed was the Beck-Chevalley condition. The dual version for pretoposes was proved by \textit{M. W. Zawadowski} [``Descent and duality'', Ann. Pure Appl. Logic 71, No. 2, 131-188 (1995; Zbl 0815.18003)] and interpreted as a definability result for coherent logic. A different, model-theoretic proof, was given later by \textit{D. Ballard} and \textit{W. Boshuk} [J. Symb. Log. 63, No. 2, 372-378 (1998; Zbl 0908.18001)]. The paper being reviewed provides yet another proof of this result which claims to be simpler than the already existing proofs. The proof relies basically on the Grothendieck construction of filtered inverse limits of toposes [\textit{M. Artin, A. Grothendi…
Mathematics Subject Classification ID: 18B25
Mathematics Subject Classification ID: 03G30
zbMATH DE Number: 1384156
zbMATH Keywords: lax pullbacks
zbMATH Keywords: topoi
zbMATH Keywords: coherent theories
zbMATH Keywords: lax descent
Wikidata QID: Q122951570
reviewed by: Marta Bunge
MaRDI profile type: Publication
full work available at URL: s0022-4049(98)00118-2
OpenAlex ID: W2031009410
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description: scientific article; zbMATH DE number 1384156