Re: Topos theory for spaces of connected components

[Marta Bunge <bunge@math.mcgill.ca>]

Dear Steve,

This is response to your message reproduced below. 

I am aware of Johnstone’s results on the lower bagdomain. However, both the symmetric monad M and the lower bagdomain monad B_L on BTop_S are “on the same side” as the lower power locale monad P_L on Loc_S, and the latter is the localic reflection for both. The upper power locale monad P_U on Loc_S is “on the other side”, in a sense that is explained in my ‘Pitts monads paper”.In it I deduce  effective lax descent theorems in a general setting of what I call "Pitts KZ-monads" and "Pitts co-KZ-monads" on a “2-category of spaces”. 

In the case of M on BTop_S, it is the S-essential surjective geometric morphisms that are shown to be of lax effective descent (a result originally due to Andy Pitts). In the case of P_L on Loc-S it is the open surjections of  locales that are shown to be of lax effective descent (a result originally  due to Joyal-Tierney). Both M and P_L are instances of "Pitts KZ-monads". Now,  P_U on Loc_S is instead an instance of a" Pitts co-KZ-monad" and the result recovered from my general setting is that proper surjections of locales are of effective lax descent (a result originally due to Jaapie Vermeulen). What I seek is a co-KZ-monad N (or perhaps B_U) on BTop_S for which my  general theorem would give me that relatively tidy surjections of toposes are of effective lax descent (a result due to I. Moerdijk and J.C.C.Vermeulen). 

In my Pitts paper there is another consequence of the general theorem proved therein and it is that coherent surjections between coherent toposes are of effective lax descent (a result proven by different methods and by several people,  such asM. Zawadowsky 1995, D.Ballard and W.Boshuck 1998, and I.Moerdijk and J.C.C.Vermeulen 1994,thus establishing a conjecture of Pitts 1985 (in the Cambridge Conference whose slides you have requested to Andy). It is of interest for what we are discussing to point out that the “coherent monad C” that I use therein to deduce the latter from my general theorem is a Pitts co-KZ-monad, hence on the “same side” as P_U for Loc_S.  For a coherent topos E, the coherent monad C(E) applied to it classifies pretopos morphisms E_{coh} —> S. where E_{coh} is the full subcategory of E of coherent objects with the topology of finite coverings. This theorem is perhaps all I can get in my setting when searching for the still elusive N or B_U but I have not given up yet. 

Also in my 2015 Pitts paper there are characterizations of the algebras for  a Pitts KZ-monad M (dually for a Pitts co-KZ-monad N) as the "stably M-complete objects" ("stably N-complete objects"), where the former is stated in  terms of pointwise left Kan extensions along M-maps, and the latter in terms of pointwise right Kan extensions along N-maps. These notions owe much to the work of M, Escardo, in particular to his 1998 "Properly injective spaces and function spaces”. 

I will say more when i know more myself. Thanks very much for your pointers. I will most certainly look into them even if I do not at the moment think  they are what I need. 


Best regards,
Marta

________________________________________________

From: Steve Vickers <s.j.vickers@cs.bham.ac.uk>
Sent: February 5, 2018 9:03 AM
To: martabunge@hotmail.com
Cc: categories@mta.ca
Subject: Re: categories: Topos theory for spaces of connected components
 
Dear Marta,

Johnstone showed that B_L(X) is a partial product of X against the "generic  local homeomorphism", a geometric morphism p from the classifier of pointed objects to the object classifier. A point of B_L(X) is a family of points  of X, indexed by elements of a set.

He also proposed other partial products, for example those against the generic entire map, which goes to the classifier for Boolean algebras from the classifier of Boolean algebras equipped with prime filter. Wouldn't that be  your B_U? A point would be a family of points of X, indexed by points of a  Stone space.

Steve.


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