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• CommentRowNumber1.
• CommentAuthorUrs
• CommentTimeJul 6th 2011

have added to Topos in the section on limits of toposes the description of the pullback of toposes by pushout of their sites of definition.

1. I have a question about colimits in $\mathbf{Topos}$.

A typical topos looks like $\mathbf{Set}$, and is therefore a large category. So $\mathbf{Topos}$ is presumably a Very large category. So we might hope that $\mathbf{Topos}$ has all large colimits. But Moerdijk only states that $\mathbf{Topos}$ has all small colimits. Of course Moerdijk might be using some different set theory conventions.

So exactly how large can colimits in $\mathbf{Topos}$ be?

2. With logical morphisms it will definitely have large colimits, because the category is then algebraic. I am not sure about the case of geometric morphisms.
• CommentRowNumber4.
• CommentAuthorMike Shulman
• CommentTimeOct 5th 2018

The page Topos is about the geometric case. It’s true that $Topos$ is set-theoretically a very large 2-category, but its sub-2-category of Grothendieck toposes (which is the one that Moerdijk is talking about) is essentially only large (though not locally small), because it is equivalent to a 2-category whose objects are small sites. So it’s not reasonable to expect it to have large colimits.

The 2-category of elementary toposes and geometric morphisms is quite ill-behaved in general; it does have some limits and colimits, but they are arguably somewhat accidental. To get good behavior you generally have to restrict to a slice category of bounded geometric morphisms over a fixed base topos, in which case things look very much again like the Grothendieck case.

3. Thanks! I was thinking purely about the case of Grothendieck toposes and geometric morphisms, but I hadn’t realised that sites had to be small in the definition of Grothendieck topos. That make things work out much more easily.