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/ ]
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/ ]
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
[For admin and other information see: http://www.mta.ca/~cat-dist/ ]
participants (3)
-
Bas Spitters -
Prof. Peter Johnstone -
Steve Vickers