Re: Reference on coverages

Re: Reference on coverages

[Prof. Peter Johnstone]

Insofar as this is a well-posed question, the answer seems to me to be
in Proposition C1.3.15 of the Elephant. (It isn't well-posed because
the subobject classifier of any topos is an object of the topos, not
something external to it.)

Peter Johnstone
---------------
On Thu, 20 Aug 2009, Bas Spitters wrote:

> The following is well-known:
> Given a posite we can construct the corresponding locale and the corresponding
> Grothendieck topos.
> Theorem: The locale is the subobject classifier of the topos.
>
> However, I fail to find a reference for this fact.
>
> Please help.
>
> Bas
>

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Re: Reference on coverages

[Steve Vickers]

Dear Bas,

I'm fairly sure this is in Joyal and Tierney (though I haven't got it
in front of me). I admit I couldn't find it in a quick look through
the Elephant. If I'm wrong about Joyal and Tierney then perhaps it's
in "Topos Theory".

If f: E -> F is a localic geometric morphism, then it is described by
f_*(Omega_E), an internal frame in F. E is the category of internal
sheaves in F for this frame. (More generally, f_*(Omega_E) is used in
constructing the hyperconnected-localic factorization of f.)

In the case where F is Sets and E is localic, then f_*(X) is the set
of global elements of any object X of E. Hence the external frame is
the set of global elements of the subobject classifier of E.

Regards,

Steve.


On 20 Aug 2009, at 21:10, Bas Spitters wrote:

> The following is well-known:
> Given a posite we can construct the corresponding locale and the
> corresponding
> Grothendieck topos.
> Theorem: The locale is the subobject classifier of the topos.
>
> However, I fail to find a reference for this fact.
>
> Please help.
>
> Bas
>



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

Reference on coverages

[Bas Spitters]

The following is well-known:
Given a posite we can construct the corresponding locale and the corresponding
Grothendieck topos.
Theorem: The locale is the subobject classifier of the topos.

However, I fail to find a reference for this fact.

Please help.

Bas


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