Re: Topos theory for spaces of connected components

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

Dear Steve,

Curiously enough, what you describe about how to get a point of MX (a cosheaf, or distribution) is precisely what did in 1990.  I communicated it privately to Lawvere at Como 1990, as he was the one who had left the question as an open one. My method to do that was that of "forcing topologies" (Tierney)and not unlike what is done in the Joyal-Tierney paper except for forgetting the lex part. Unfortunately, when I sent a paper "Cosheaves and distributions on toposes" containing this result to Peter Freyd for the JPAA, whoever refereed it rejected it without giving a reason, but I know that it was my fault as the paper was not too clearly written. I subsequently sent it to Algebra Universalis where it did appear (same title and alas, still not too clearly written) but very much later (1995).

That sabbatical year 1995-96 I was spent at the Universita di Genova where (not coincidentally) Aurelio (Carboni) had just taken a job away from Milan. Aurelio immediately understood my construction, but thought that it would  be "wiser" to set it in algebraic terms. This resulted in our joint paper "The symmetric topos", which this time it did appear in JPAA (1995). We did  more than that in that paper, namely to extend it to a KZ-monad and characterize its algebras. I therefore abandoned the fibrational point of view which, as you say, found its way again in my work with Jonathon (Funk) as the  complete spreads with a locally connected domain. However, just this morning (and before reading what you just wrote) I was thinking of using the fibrational approach again for constructing the topos NX corresponding to the upper power locale, just as the topos MX corresponds to the lower power locale. 

It seems to me now that modulo some differences this is what you are trying  to do yourself. Of course it would be "wiser" as Aurelio would have said, to do it "algebraically" and this is what I did for the coherent monad in my "Pitts paper" (2015). So I am pursuing that line as well. I will read the  rest of your message later and maybe respond to your question about it privately. It may take a few days as other things are interfering with my work  at present. 

Many thanks for your remarks. 

All the best,
Marta     

----- Original Message -----
From: "Steve Vickers" <s.j.vickers@cs.bham.ac.uk>
To: bunge@math.mcgill.ca
Cc: categories@mta.ca
Sent: Tuesday, February 6, 2018 4:19:50 AM
Subject: Re: categories: Topos theory for spaces of connected components

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,


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