Re: Categorical problems

["Eduardo J. Dubuc" <edubuc@dm.uba.ar>]

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 ?).


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