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.
-----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.
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
participants (3)
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Barney Hilken -
Prof. Peter Johnstone -
Steve Lack