Re: Famous unsolved problems in ordinary category theory

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

Dear Hasse,

The presheaves that Townsend and I used are on the category Loc of
locales. The fact that Loc is large may be seen as a problem, but
another illegitimacy is the way a presheaf is a functor to Sets. This
means, for instance, that for a representable presheaf y(X), where X is
a locale, we take y(X)(W) = Loc(W,X) to be a _set_ for any pair of
locales X and W, and that is foundationally tendentious. Whatever kind
of collection Loc(W,X) is (if W is locally compact then we can take it
to be another locale, but otherwise not), the ability to extract a "set
of points" from it is sensitive to the foundations.

We tried to be foundationally conservative in what we did with the
presheaves, and you can see come remarks on this in the conclusions of
our paper. (I should stress that we did not claim to have embedded Loc
in a CCC, and we tried not to make use of any particular categorical
properties of Presh(Loc).) Insofar as the representable presheaves y(X)
can be acceptable, then so too are their exponentials y(Y)^y(X), since
y(Y)^y(X)(W) = Loc(WxX,Y). What we showed is that then the exponential
y($)^(y($)^y(X)) also exists (where $ = the Sierpinski locale), and in
fact is representable of the form y(PP(X)) where PP(X) is the "double
powerlocale" on X. Thus PP(X) has a claim to be thought of as $^($^X)
even when X is not exponentiable (locally compact). PP is a
foundationally robust construction, available in both topos-valid locale
theory and predicative formal topology.

Regards,

Steve Vickers.

Hasse Riemann wrote:
>> Another is how to embed the category of locales in a CCC WITHOUT
>> using illegitimate presheaves (Vickers and Townsend) or the axiom
>> of collection (Heckmann).
>
> I don't follow to the end here.
> Why should presheaves be illegitimate?


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