Categorical problems

[Ross Street <street@ics.mq.edu.au>]

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Walter Tholen's recent message reminded me of a conjecture. Perhaps
we have been too shy about stating our conjectures because, even
among mathematicians, they may have seemed too technical.

I seem to remember Peter Freyd saying once that the problem in
category theory of proving sets were small (to find adjoints to
functors for example) was analogous to finding numerical bounds in
mathematical analysis. Surely by now, there are as many people who
understand what a sheaf is as understand what the Riemann Hypothesis
asserts (for example, local to global versus analytic continuation).

So here is a problem I came up with in the 1970s. As with Fermat's
Last Theorem, I don't particularly remember having any application
for it. However, similar solved problems were used by Rosebrugh-Wood
to characterize the category of sets in terms of adjoint strings
involving the Yoneda embedding. By locally small I mean having homs
in a chosen category Set of small sets.

Problem. Suppose A is a locally small site whose category E of Set-
valued sheaves is also locally small. Is E a topos?

== Ross

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