Re: Topos theory for spaces of connected components

[Steve Vickers <s.j.vickers@cs.bham.ac.uk>]

Dear Marta,

Here's my thinking on connected components.

For M, the paradigm example for how to get a point of MX (a cosheaf, or distribution) is to take locally connected space Y with map p: Y -> X, and then to each sheaf U over X assign the set of connected components of p*U. This gives a covariant functor from SX to Set, and it preserves colimits. If X is an ungeneralized space, then it suffices to do that for opens U, and the extension to sheaves follows. Your theory of complete spreads shows that that paradigm example is in fact general.

The extreme case of p is when X is itself locally connected and we can take p  to be the identity. The corresponding cosheaf is terminal in a strong sense: as global point of MX it provides a right adjoint to the map MX -> 1. The unit of the adjunction provides a unique morphism from the generic cosheaf to  the terminal one.

If X is exponentiable, then (always? In favourable cases?) the cosheaf as described above can be got by taking points for a map R^X -> R, where R is (following your notation) the object classifier. This points out Lawvere's analogy with integration, where R would be the real line. Then just as Riesz picks out the linear functionals as the distributions, we are interested in the  colimit-preserving ones.

In the above account, the role of local connectedness is to ensure that the connected components genuinely do form a set, a discrete space. What happens if we look for a Stone space instead? Here is my conjecture.

1. For ungeneralized X we should be looking for a Stone space of connected components of p*U for each _closed_ U. Y will need a suitable condition (strongly compact?) as analogue of local compactness. (By Stone duality that could also be expressed by assigning (covariantly) a Boolean algebra to each open.)

2. Noting that a closed embedding is fibrewise Stone, that assignment will extend to U an arbitrary fibrewise Stone (entire) bundle over X - that is to say, by Stone duality and contravariantly, a sheaf of Boolean algebras.

3. For generalized X that will provide our Stone notion of cosheaf. The assignment from entire bundles to Stone spaces should preserve finite colimits and cofiltered limits. There's an obvious technical hurdle of how to express that directly in terms of sheaves instead of entire bundles.

4. If X is exponentiable then this time, by Stone duality, we are looking for maps [BA]^X -> [BA] where [BA] is the classifier for Boolean algebras. They must preserve filtered colimits (automatic for maps) and finite limits. NX  would exist for arbitrary X, and classify those maps. 

Obviously there's lots to go wrong there, but do you think your coherent monad fits any of those points for coherent X?

By the way, although I haven't mention the effective lax descent and relatively tidy maps, I am interested in them. They are connected with stable compactness and Priestley duality.

All the best,

Steve.

> On 5 Feb 2018, at 18:07, bunge@math.mcgill.ca wrote:
> 
> 
> 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
> 

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