A la recherche de N

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

Dear fellow categorists, 


Preamble. 

I have a question that is the key to getting the (co)KZ-monad Tau on the opposite R of the 2-category bounded S-toposes “just as” Sigma was defined in Bunge-Carboni “The symmetric topos” in 1995. In turn, just as taking opposites to Sigma gives the KZ-monad M on BTop_S, pursued in several papers including the Bunge-Funk 2006 book, Singular Coverings of Toposes.  In my “Pitts monads and a lax descent theorem” (2015)  I used M to derive that S-essental surjections of toposes  are of effective descent, (a result due to Pitts (1986), along with other known and unknown (lax)descent theorems, moreover in a unified manner. The goal here is to do likewise by taking opposites to Tau, thus giving a (co)-KZ monad N on BTop_S, to be used among other things, to derive that relatively tidy surjections of toposes are of effective lax descent, a result due to Moerdijk and Vermeulen (2000).  In the Bunge-Carboni paper, Sigma was obtained as the left adjoint to the forgetful R—> A, where A was the 2-category of locally presentable categories and cocontinuous functors as morphisms. It involved a lex completion at the level of the sites. The key then was simply to know that coinverters in R and in A both existed and that  those in R could be calculated in A. We took the word of Max Kelly and Andy Pitts that this was indeed the case. Dually, Tau (if it existed) should be the left adjoint to the forgetful R—> B, where B now is the 2-category of locally presentable categories with lex and filtered colimit preserving functors as morphisms. 

The Question. 

Let B be the 2-category of locally presentable categories with lex and filtered colimit preserving functors as morphisms. Let R be the opposite of the  2-category BTop_S of bounded S-toposes, geometric morphisms and 2-cells. Do coinverters in B exist? If so,  are those in R calculated just as in B? If not, is there a known class of toposes, strictly larger than that of the coherent toposes, for which this is the case?

Further comments.

The locales version of an answer to my question is in the Johnstone-Vickers  paper “Preframe presentations present’ (1991) but, just as  with Carboni we did not look at suplattices for an inspiration of how to get Sigma, I do not want to do likewise for Tau looking at locale preframes,  as the transition from locales to toposes is not perfect. Let me point out   howevet that the case of N restricted to coherent toposes is easy in that  setting, as the sites of coherent toposes are categories with finite limits already, so that the frame completion involves only adding finite colimits, unlike the frame completion of a preframe which involves adding both finite limits and filtered colimits (plus the distributive aspect).  I recently learnt that also Steve Vickers and Christopher Townsend have been looking  for this N for a long time and after learning about the symmetric monad (Sigma and M). Steve, who wrote to categories about it last week, is actually   looking for the geometric counterpart of (the points of) an N(E) for E a topos, just as the complete spreads with a locally connected domain are the  counterpart of  the Lawvere distributions on toposes, that is, of (the points)  of M(E). 

Conclusion.

Any concrete pointers to an answer will be appreciated. I am aware that there would be other ways to tackle this completion-cocompletion matter (bicompletions, distributive laws) if it were just a matter of getting N, but not  so for my current purpose as I explained above. 

Best wishes,
Marta












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