Re: Grothendieck toposes

[wlawvere <wlawvere@buffalo.edu>=0A=]

Dear friends and colleagues,

In Spring 1981, near a lavender field in Southern France,
Alexander Grothendieck greeted me at the door of his home.
He wasted no time and immediately put the question:

'What is the relationship between the two uses of the term 'topos'?'

This led to a very interesting discussion.The first thing that
was established as a basis was that SGA4 never defined 'topos',
but rather spoke always of 'U-topos',  where U was a certain
kind of model of set theory. All the categories so arising have
common features, such as cartesian closure, and the U itself can
be construed as such a category. (TAC Reprints no. 11).

Thus we arrived at the notion of 'U-topos' as a special geometric
morphism E →U  of 'elementary' toposes. Grothendieck's
general method of relativization suggests the usefulness
of a general topos as a codomain or base U. (see Giraud, SLN 274).
But to focus more specifically on the original case, various special
properties of the base U could also be considered:
Booleanness (note for example, that Booleanness distinguishes
algebraic points among algebraic figures)
Axiom of choice;
Lack of measurable cardinals; et cetera.

One of the many topics we discussed was the
'Medaille de Chocolat' exercise in SGA4, and its basic importance
for understanding applications of topos theory: the gros and
petit sheaves of an object point out that there should be a
qualitative distinction between a topos of SPACES and a topos
of set-valued sheaves on a generalized space. I believe that
considerable progress is now being made on the characterization
of 'gros' toposes under the name of Cohesion. Grothendieck made
a big step towards  the characterization of 'petit' under the name
of  'etendu'  (sometimes known as 'locally localic'). Concerning
Grothendieck's most famous contribution, the 'petit etale' topos,
what are it's distinguishing properties as a topos?

We also discussed the Grauert direct image theorem as a
relativization of the Cartan-Serre theorem. It is important to
note that Grothendieck's work was not limited to the Weil
conjectures but, for example, involved around 1960 several
categories related to complex analysis which were perhaps
part of his inspiration for the notion of topos.


Separation?
Actually, separation has been one of the main sources of confusion.
I wish that someone with internet confidence would correct the
Wikipedia article that claims that pre-1970 toposes were about
geometry, but that post-1970 toposes were about logic. Certainly,
that discourages students from studying either.
Omitted was the fact that logic has always been used to sharpen the
study of geometry; in the last 50 years we have been able to make
this relation more explicit, with the help of categories.

Of course, separating a certain kind of object from a certain kind
of map would be basic 'grammar'.
But we cannot separate the legacy of Grothendieck from the
inspiration it gives to the continuing development of topos theory.

Best wishes
Bill Lawvere





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