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1. • CommentRowNumber1.
   • CommentAuthorUrs
   • CommentTimeJul 6th 2011
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   Author: Urs
   Format: MarkdownItexhave added to [[Topos]] in the section [on limits of toposes](http://ncatlab.org/nlab/show/Topos#Limits) the description of the pullback of toposes by pushout of their sites of definition.

   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.

2. • CommentRowNumber2.
   • CommentAuthorOscar_Cunningham
   • CommentTimeOct 5th 2018
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   Author: Oscar_Cunningham
   Format: MarkdownItexI 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 <a href="https://ncatlab.org/nlab/show/Topos#Moerdijk">Moerdijk</a> 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?

   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?

3. • CommentRowNumber3.
   • CommentAuthorRichard Williamson
   • CommentTimeOct 5th 2018
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   Author: Richard Williamson
   Format: TextWith logical morphisms it will definitely have large colimits, because the category is then algebraic. I am not sure about the case of geometric morphisms.

   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.
4. • CommentRowNumber4.
   • CommentAuthorMike Shulman
   • CommentTimeOct 5th 2018
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   Author: Mike Shulman
   Format: MarkdownItexThe 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.

   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.

5. • CommentRowNumber5.
   • CommentAuthorOscar_Cunningham
   • CommentTimeOct 5th 2018
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   Author: Oscar_Cunningham
   Format: MarkdownItexThanks! 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.

   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.