Giraud_Elementary_?

Giraud_Elementary_?

[Eduardo Julio Dubuc]

Hi, in this posting I will use the terminology used by most people in
the list.

There are Grothendieck, Giraud and Elementary (Lawvere-Tierney) topos.

Grothendieck are Giraud and Elementary, my question is:

Are Elementary Giraud topos which are not Grothendieck ?

Examples ?

greetings  e.d.


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Re: Giraud_Elementary_?

[Thomas Streicher]

On Mon, Nov 07, 2016 at 04:03:18PM -0500, Eduardo Julio Dubuc wrote:
> Hi, in this posting I will use the terminology used by most people in
> the list.
>
> There are Grothendieck, Giraud and Elementary (Lawvere-Tierney) topos.
>
> Grothendieck are Giraud and Elementary, my question is:
>
> Are Elementary Giraud topos which are not Grothendieck ?

But Grothendieck and Giraud toposes are the same. In the Elephant one
can even find a relative Giraud Theorem.

Thomas


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Re: Giraud_Elementary_?

[Eduardo Julio Dubuc]

On 11/8/16 05:03, Thomas Streicher wrote:
> On Mon, Nov 07, 2016 at 04:03:18PM -0500, Eduardo Julio Dubuc wrote:
>> Hi, in this posting I will use the terminology used by most people in
>> the list.
>>
>> There are Grothendieck, Giraud and Elementary (Lawvere-Tierney) topos.
>>
>> Grothendieck are Giraud and Elementary, my question is:
>>
>> Are Elementary Giraud topos which are not Grothendieck ?
>
> But Grothendieck and Giraud toposes are the same. In the Elephant one
> can even find a relative Giraud Theorem.
>
> Thomas
>

By Giraud topos I mean all the assumptions in Giraud's theorem, exept a
small set of generators. What Grothendieck call "faux topos".

See SGA4 Exposse IV
Theoreme 1.2 (Giraud's theorem) and Example 2.8 (faux topos).

best  e.d.


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Re: Giraud_Elementary_?

[Eduardo Julio Dubuc]

By Giraud topos I mean all the assumptions in Giraud's theorem, exept a
small set of generators. What Grothendieck call "faux topos".

See SGA4 Exposse IV
Theoreme 1.2 (Giraud's theorem) and Example 2.8 (faux topos).

best  e.d.

I guess I was wrong when I thought that "Giraud Topos" was established
terminology in the cat-list.




On 11/8/16 09:59, Daniil Frumin wrote:
> What is actually a Giraud topos? I cannot find a reference for this on
> the internet.
>
> On Mon, Nov 7, 2016 at 10:03 PM, Eduardo Julio Dubuc <edubuc@dm.uba.ar
> <mailto:edubuc@dm.uba.ar>> wrote:
>
>     Hi, in this posting I will use the terminology used by most people in
>     the list.
>
>     There are Grothendieck, Giraud and Elementary (Lawvere-Tierney) topos.
>
>     Grothendieck are Giraud and Elementary, my question is:
>
>     Are Elementary Giraud topos which are not Grothendieck ?
>
>     Examples ?
>
>     greetings  e.d.
>
>
>     [For admin and other information see: http://www.mta.ca/~cat-dist/ ]
>
>



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Re: Giraud_Elementary_?

[henry]

Dear Eduardo

Unless you are using a different statement of the Giraud's theorem than
the one I have in mind, they are I think considerably more often called
$\infty$-pretopos (like in the Elephant) or infinitary pretopos (like in
the nLab) to avoid any confusion with an infinity categorical notion. I
don't think I have ever encountered a different terminology (but I do like
'Giraud topos').

Regarding the example you are looking for, unless I'm missing something,
the example 2.8 in SGA that you mentioned (the category sets endowed with
smooth action of a large group) is also an elementary topos:  sub-object
classifier, exponential and power object are constructed exactly in the
case of an ordinary group action topos and only involve a small quotient
of the large group. So it answer you question.

Bests,
Simon

> By Giraud topos I mean all the assumptions in Giraud's theorem, exept a
> small set of generators. What Grothendieck call "faux topos".
>
> See SGA4 Exposse IV
> Theoreme 1.2 (Giraud's theorem) and Example 2.8 (faux topos).
>
> best  e.d.
>
> I guess I was wrong when I thought that "Giraud Topos" was established
> terminology in the cat-list.
>
>

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