Re: Grothendieck toposes

[Thomas Streicher <streicher@mathematik.tu-darmstadt.de>]

As to the question brought up by Steve I want to remark that
Grothendieck always spoke about U-topos for some (Grothendieck)
universe U. To write simply Set instead of U is covenient but slightly
misleading if one takes logical foundations seriously since what after
all is this Set? One shouldn't forget that ZFC has many models even
when one adds the axiom that every set is element of a Grothendieck
universe (it has a countable model by downward Loewenheim-Skolem).

If one uses an extended set-theoretic foundation as Grothendieck did
Set is just a name for a generic Grothendieck universe.

With the advent of elementary topos theory one wanted to forget about
set theory since one thought that category theory provide its own
foundation via elementary toposes. This certainly makes sense but what
then is Set? Well, one may choose some (unspecified) base topos SS and
consider categories relative to SS as Grothendieck fibrations over SS.
The role of "Set" is then taken by the fundamental ("codomain") fibration
P_SS = cod : SS^2 -> SS (where 2 is the ordinal 2). From this relative
point of view Grothendieck toposes over SS correspond to bounded
geometric morphisms to SS as worked out in detail in Johnstone's 1977 book.

But, of course, there may be many non isomorphic g.m.s from EE to SS.
However, in Top/SS there is a (kind of) terminal object, the identity
g.m. on SS. As explained in Moens's 1982 Thesis g.m.s to SS correspond
to cocomplete locally small fibrations of toposes over SS (he assumed
that the internal sums were stable and disjoint which 6 years later
was shown by Jibladze to be the case for all cocomplete fibered toposes.

But if one has a Grothendieck topos EE over SS the internal language
of SS doesn't allow one to speak about EE in all relevant respects.
In particular, one cannot quantify over the objects of EE within the
internal language of SS.

However, one may "blow up" SS so that one can. It is an old
observation by Benabou that *split* fibrations over SS correspond to
categories internal to presheaves over SS (for a large enough "Set"). This,
however, is not possible for non-split fibrations like P_SS. Different
ways of overcoming this problem have been found by Awodey, Butz,
Simpson and myself "Relating first-order set theories, toposes and
categories of classes" (APAL 2014) and in an unpublished paper by
Mike Shulman arXiv:1004.3802.

The restrictions of the internal language of the base topos
w.r.t. speaking about a fibration over it can be overcome when one
admits universes in the base topos. These universes are less
set-theoretic than Grothendieck's ones but play a similar role.

Thomas


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