Re: Grothendieck toposes

Re: Grothendieck toposes

[wlawvere]

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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Re: Re: Grothendieck toposes

[Steve Vickers]

Dear Bill,

I'd really like to understand these issues about gros and petit toposes better.

My own current direction with arithmetic universes is - I believe - a reasonable one to try in respect to geometric theories and classifying toposes, but it loses sight of gros toposes and synthetic approaches, which, after all,  were part of the founding ideas of toposes. I need to know better what it is that I risk losing.

However, my attempts to understand gros toposes run into difficulties with geometricity. This is illustrated by the Zariski topos for algebras over a field k. If k is R or C then the geometric methodology expects it and its algebras to be locales (and non-discrete), not sets. But does the polynomial ring R[x] exist localically? The degree of a polynomial looks like being part of the geometric structure, and removing leading zeros (e.g. after a subtraction) is not continuous.

Do you know if anyone has investigated a localic form of these methods in algebraic geometry?

All the best,

Steve.

p.s. I like to think I'm following Grothendieck's insights, but the truth is  I understand only a tiny fraction of them.

> On 6 Nov 2016, at 15:41, wlawvere =0A= <wlawvere@buffalo.edu> wrote:
> 
> 
> 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

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