-----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.