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From: "Prof. Peter Johnstone" <P.T.Johnstone@dpmms.cam.ac.uk>
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Subject: Re: Pullback & coproduct of toposes
Date: Tue, 29 Jun 2004 18:01:49 +0100 (BST)
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On Tue, 22 Jun 2004, Steve Lack wrote:

> >I don't have access to a decent library at the moment, and I can't
> >afford a copy of the Elephant myself, so can anyone let me know:
> >
> >In the category of toposes and geometric morphisms, under what
> >conditions is coproduct stable under pullback?
> >
>
> Answer: always. Let f:E-->S+S' be a morphism of toposes. Identify S+S'
> with the product of the categories S and S'. Then in S+S' the terminal
> object (1,1) is a coproduct (1,0)+(0,1). Now apply the inverse image
> functor f* to obtain a decomposition 1=X_1+X_2 of the terminal object 1
> in E. By extensivity of E, then, the category E is equivalent to the
> product E/X_1 x E/X_2; in other words, the topos E is the coproduct
> of the toposes E/X_1 and E/X_2. (Where E/X_1 and E/X_2 are of course
> the pullbacks along f of the injections S-->S+S' and S'-->S+S'.)
>
> This argument is contained in
>
> Marta Bunge & Stephen Lack, Van Kampen theorems for toposes, Adv. Math.
> 179:291-317, 2003.
>
> where it is seen as part of the fact that the 2-category of toposes
> is extensive.
>
... and it is (of course) in the Elephant: page 402, remark following
Lemma B3.4.1.

Peter Johnstone