RE: Pullback & coproduct of toposes

["Steve Lack" <s.lack@uws.edu.au>]

>-----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.