You're reading the public-facing archive of the Category Theory Zulip server.
To join the server you need an invite. Anybody can get an invite by contacting Matteo Capucci at name dot surname at gmail dot com.
For all things related to this archive refer to the same person.
[You can replace "Grothendieck toposes" with "toposes bounded over S" for your favourite elementary topos S if you like.]
The initial topos (the degenerate one) is a strict 2-initial object. One can construct coproducts of toposes as the product of their underlying categories. The boundedness/Grothendieck restriction is enough to guarantee the existence of pullbacks, and the pullback of the coproduct inclusions is the initial category, so coproducts are disjoint (two toposes are complementary clopen subtoposes in their coproduct). We're in good shape, so the question is, is this 2-category extensive (in the most straightforward sense that one can generalize extensivity to this setting)? Has anyone written about this?
A related question whose answer is not necessarily determined by the above: are unions of subtoposes preserved by pullbacks in this topos? They are stable by pullback along inclusions, but what about surjections?
After showing that the product of logoses is the coproduct of toposes (B3.4.1), Sketches of an Elephant remarks
Also, coproducts are disjoint and stable under pullback; for we can identify and with the complementary open subtoposes and of the coproduct, and any geometric morphism induces a corresponding decomposition of its domain.
"Coproducts are disjoint and pullback-stable" is one definition of extensive for 1-categories. I don't know offhand whether that is equivalent to something about slice categories as in the 1-categorical case, is that more what you had in mind?
That is what I had in mind, thanks; I suppose the term extensive for this property was popularised after the Elephant was published. I'm still interested in unions of subtoposes though!
I would expect that you can construct the union of subtoposes by taking their coproduct and then the (surjection, inclusion) factorization. In which case, pullback-stability of unions should follow from pullback-stability of coproducts and surjections.
Surjections are not stable under pullback(!)
Some classes of surjection are stable, but which of the properties ensuring stability will be inherited by the surjection part of the universal morphism from the coproduct is not obvious (some of them might be checkable manually though)