Categorical problems

Categorical problems

[Ross Street]

[From moderator: Resent with apologies to those who received an empty
message body...]

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

[For admin and other information see: http://www.mta.ca/~cat-dist/ ]

Re: Categorical problems

[Prof. Peter Johnstone]

On Mon, 8 Jun 2009, Eduardo J. Dubuc wrote:

> Ross Street wrote:
>
>> Problem. Suppose A is a locally small site whose category E of Set-
>> valued sheaves is also locally small. Is E a topos? (see (*) below)
>
> This is one of (probably) many problems of Girau topoi [satisfy all
> conditions
> in Girau's Theorem exept (may be) the set of generators] which are not known
> to be a topos.
>
> (*) It seems Not: Take a Girau (really faux but locally small) topos E, with
> the canonical topology. Then the topos of sheaves should be E again, which is
> not a topos (am I missing something ?).
>
I presume that Ross was using the word "topos" to mean "elementary topos".
But in any case, Eduardo was missing something: the proof that, if E is
an \infty-pretopos (my preferred name for what he calls a "Girau(d)
topos"), then every canonical sheaf on E is representable, requires the
existence of a generating set (see C2.2.7 in the Elephant). For a
counterexample in the absence of generators, let G be the "large" group
of all functions from the ordinals to {0,1} having finite support, the
group operation being pointwise addition mod 2 (or, if you prefer,
the group of finite subsets of the ordinals under symmetric difference),
and let E be the (elementary) topos of G-sets. For each ordinal \alpha,
let A_\alpha be the set {0,1} with G acting via its \alpha-th factor;
then any G-set admits morphisms into only a set of the A_\alpha, from
which it follows that the coproduct of all the A_\alpha exists as a
(set-valued) canonical sheaf on E, though it clearly isn't a set.
Moreover, this coproduct admits a proper class of maps to itself, so
the category of sheaves on E isn't locally small; hence it doesn't
violate Ross's conjecture.

Peter Johnstone



[For admin and other information see: http://www.mta.ca/~cat-dist/ ]

Re: Categorical problems

[Eduardo J. Dubuc]

Ross Street wrote:

 > Problem. Suppose A is a locally small site whose category E of Set-
 > valued sheaves is also locally small. Is E a topos? (see (*) below)

This is one of (probably) many problems of Girau topoi [satisfy all conditions
in Girau's Theorem exept (may be) the set of generators] which are not known
to be a topos.

Another, the Etale "topos" in the sense of Joyal's axiomatic theory of etal
maps (which is even a subcategory of a topos).

Another (solved), to show the existence of colimits in the category of topoi,
the only hard part is to get the generators.

Concerning the other thread (not Ross question)

 > > My question is, What would be candidates for the Fundamental Theorem
 > > of Category Theory?
 > >
 > > Yoneda Lemma comes to my mind. What do you think?


Of course, Yoneda Lemma, at the birth of category theory, is the fundamental
result that makes of category theory something more than a convenient language.

Related to this, the definition of category should include small hom sets, and
categories with large hom sets should be called "illegitimate" (in the manner
of the definition of topoi, which include generators, the others being
illegitimate or "faux" in Grothendieck's terminology).

(*) It seems Not: Take a Girau (really faux but locally small) topos E, with
the canonical topology. Then the topos of sheaves should be E again, which is
not a topos (am I missing something ?).


[For admin and other information see: http://www.mta.ca/~cat-dist/ ]

Categorical problems

[Ross Street]

[-- Attachment #1: Type: multipart/alternative, Size: 1169 bytes --]