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June 2015 Pitts monads and a lax descent theorem
Marta Bunge
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Tbilisi Math. J. 8(1): 1-29 (June 2015). DOI: 10.1515/tmj-2015-0001


A theorem of A.M.Pitts (1986) states that essential surjections of toposes bounded over a base topos $\mathscr{S}$ are of effective lax descent. The symmetric monad $\mathscr{M}$ on the 2-category of toposes bounded over $\mathscr{S}$ is a KZ-monad (Bunge-Carboni 1995) and the $\mathscr{M}$-maps are precisely the $\mathscr{S}$-essential geometric morphisms (Bunge-Funk 2006). These last two results led me to conjecture1 and then prove2 the general lax descent theorem that is the subject matter of this paper. By a ‘Pitts KZ-monad’ on a 2-category $\mathscr{K}$ it is meant here a locally fully faithful equivariant KZ-monad $\mathscr{M}$ on $\mathscr{K}$ that is required to satisfy an analogue of Pitts' theorem on bicomma squares along essential geometric morphisms. The main result of this paper states that, for a Pitts KZ-monad $\mathscr{M}$ on a 2-category $\mathscr{K}$ (‘of spaces’), every surjective $\mathscr{M}$-map is of effective lax descent. There is a dual version of this theorem for a Pitts co-KZ-monad $\mathscr{N}$. These theorems have (known and new) consequences regarding (lax) descent for morphisms of toposes and locales.


Dedicated to Marco Grandis on his 70th Birthday


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Marta Bunge. "Pitts monads and a lax descent theorem." Tbilisi Math. J. 8 (1) 1 - 29, June 2015.


Received: 28 May 2014; Accepted: 18 November 2014; Published: June 2015
First available in Project Euclid: 12 June 2018

zbMATH: 1350.18008
MathSciNet: MR3314179
Digital Object Identifier: 10.1515/tmj-2015-0001

Primary: 03G
Secondary: 18C, 18D, 18F, 55R

Rights: Copyright © 2015 Tbilisi Centre for Mathematical Sciences


Vol.8 • No. 1 • June 2015
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