Pullback & coproduct of toposes

Pullback & coproduct of toposes

[Barney Hilken]

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?

Any reasonable conditions on the toposes or the morphisms would be
helpful, but the more general, the better.
Thanks,

Barney.

RE: Pullback & coproduct of toposes

[Steve Lack]

>-----Original Message-----
>From: cat-dist@mta.ca [mailto:cat-dist@mta.ca]On Behalf Of Barney Hilken
>Sent: Wednesday, 16 June 2004 1:38 AM
>To: categories@mta.ca
>Subject: categories: Pullback & coproduct of toposes
>
>
>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.

Steve Lack.

Re: Pullback & coproduct of toposes

[Prof. Peter Johnstone]

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