Re: Topos theory for spaces of connected components

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

Correction - In Conjecture 1 I mistakenly wrote "local compactness" for
"local connectedness".
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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 connectedness. (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.


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