Open Access
Translator Disclaimer
June 2015 Pitts monads and a lax descent theorem
Marta Bunge
Author Affiliations +
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


Download Citation

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

Keywords: coherent toposes , Kock-Zoeberlein monads , lax descent , Pitts' theorem , powerlocales , symmetric monad

Rights: Copyright © 2015 Tbilisi Centre for Mathematical Sciences


Vol.8 • No. 1 • June 2015
Back to Top