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1. • CommentRowNumber1.
   • CommentAuthorTodd_Trimble
   • CommentTimeSep 14th 2011
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   Author: Todd_Trimble
   Format: MarkdownItexI was wondering whether anyone had a pleasant concrete description of the category of functors $Set^{op} \to Set$ which are small colimits of representables. I've just spent the last few minutes quickly skimming through the paper by Day and Lack, Limits of small functors, but I didn't see what I'm after. Is there any chance that this thingy is a topos?

   I was wondering whether anyone had a pleasant concrete description of the category of functors $Set^{op} \to Set$ which are small colimits of representables. I’ve just spent the last few minutes quickly skimming through the paper by Day and Lack, Limits of small functors, but I didn’t see what I’m after.

   Is there any chance that this thingy is a topos?

2. • CommentRowNumber2.
   • CommentAuthorTodd_Trimble
   • CommentTimeSep 14th 2011
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   Author: Todd_Trimble
   Format: MarkdownItexJust to add a small remark: the small-coproduct completion of a Grothendieck topos $E$ is a topos. Specifically, it can be realized as a gluing construction $E \downarrow \Delta$ where $\Delta: Set \to E$ is the essentially unique left exact left adjoint. I was hoping something similar could be worked out for the small-colimit completion. I was thinking "all we needed to do" is take the "coequalizer-completion" of the small-coproduct completion. But the coequalizer completion looks potentially nasty. Still, I'm throwing it out there in case it suggests anything.

   Just to add a small remark: the small-coproduct completion of a Grothendieck topos $E$ is a topos. Specifically, it can be realized as a gluing construction $E \downarrow \Delta$ where $\Delta: Set \to E$ is the essentially unique left exact left adjoint. I was hoping something similar could be worked out for the small-colimit completion.

   I was thinking “all we needed to do” is take the “coequalizer-completion” of the small-coproduct completion. But the coequalizer completion looks potentially nasty. Still, I’m throwing it out there in case it suggests anything.

3. • CommentRowNumber3.
   • CommentAuthorTodd_Trimble
   • CommentTimeSep 16th 2011
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   Author: Todd_Trimble
   Format: MarkdownItexStill chipping away at this. The small-colimit completion of a topos $E$ seems to be the same as the exact (or ex/lex) completion of $\Sigma E$, the small-coproduct completion of $E$. It looks as though, from a result in Menni's thesis, that $(\Sigma E)_{ex}$ ought to have a generic proof (because $\Sigma E$ has finite limits; see the first full paragraph on page 55) and therefore be a topos, but there may be some tricky size considerations I haven't digested. Hey, where is Mike these days? :-)

   Still chipping away at this. The small-colimit completion of a topos $E$ seems to be the same as the exact (or ex/lex) completion of $\Sigma E$, the small-coproduct completion of $E$. It looks as though, from a result in Menni’s thesis, that $(\Sigma E)_{ex}$ ought to have a generic proof (because $\Sigma E$ has finite limits; see the first full paragraph on page 55) and therefore be a topos, but there may be some tricky size considerations I haven’t digested.

   Hey, where is Mike these days? :-)

4. • CommentRowNumber4.
   • CommentAuthorTodd_Trimble
   • CommentTimeSep 17th 2011
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   Author: Todd_Trimble
   Format: MarkdownItexWell, I seem to have an answer to my question at the end of #1. No, it's not a topos. :-( I'm getting this from what might be a pirated copy of a paper (thus I don't want to say more about it here), but the paper itself is completely trustworthy. I discovered that exact completions of presheaf toposes $Set^{C^{op}}$ are toposes only when $C$ is a groupoid (which is interesting in a negative kind of way).

   Well, I seem to have an answer to my question at the end of #1. No, it’s not a topos. :-(

   I’m getting this from what might be a pirated copy of a paper (thus I don’t want to say more about it here), but the paper itself is completely trustworthy. I discovered that exact completions of presheaf toposes $Set^{C^{op}}$ are toposes only when $C$ is a groupoid (which is interesting in a negative kind of way).

5. • CommentRowNumber5.
   • CommentAuthorMike Shulman
   • CommentTimeOct 3rd 2011
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   Author: Mike Shulman
   Format: MarkdownItexSorry I was gone when you wanted me! This is a good question. What you say about exact completions of presheaf toposes sounds vaguely familiar, but I don't recall where I might have read it. On the other hand, I believe the small-colimit completion of any finitely complete category is an infinitary pretopos; you might be interested in [these slides](http://www.math.ucsd.edu/~mshulman/papers/exsitetalk.pdf). With luck, a more detailed writeup will be forthcoming soon....

   Sorry I was gone when you wanted me! This is a good question. What you say about exact completions of presheaf toposes sounds vaguely familiar, but I don’t recall where I might have read it.

   On the other hand, I believe the small-colimit completion of any finitely complete category is an infinitary pretopos; you might be interested in these slides. With luck, a more detailed writeup will be forthcoming soon….

6. • CommentRowNumber6.
   • CommentAuthorTodd_Trimble
   • CommentTimeOct 3rd 2011
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   Author: Todd_Trimble
   Format: MarkdownItex> What you say about exact completions of presheaf toposes sounds vaguely familiar, but I don't recall where I might have read it. It's in a paper by Menni (details available if anyone wants to email me). Mike, I might have emailed you privately about stuff like the following conjecture: if you have a pullback-preserving comonad acting on a $\Pi$-$W$-pretopos (by which I mean a complete, cocomplete, lcc pretopos with initial algebras for any polynomial endofunctor), then the category of coalgebras is also a $\Pi$-$W$-pretopos. (Similar but slight more tentative conjecture, where "pullback-preserving comonad" is replaced by "pullback-preeserving modal operator", as discussed on the page [here](http://ncatlab.org/toddtrimble/published/Three+topos+theorems+in+one).) I have some incomplete notes on this stuff on my web page, [here](http://ncatlab.org/toddtrimble/published/Small+cocompletions), which has been left in a state of limbo. Mean to finish some thoughts here... I'll take a look at your slides when I get a chance.

   What you say about exact completions of presheaf toposes sounds vaguely familiar, but I don’t recall where I might have read it.

   It’s in a paper by Menni (details available if anyone wants to email me).

   Mike, I might have emailed you privately about stuff like the following conjecture: if you have a pullback-preserving comonad acting on a $\Pi$-$W$-pretopos (by which I mean a complete, cocomplete, lcc pretopos with initial algebras for any polynomial endofunctor), then the category of coalgebras is also a $\Pi$-$W$-pretopos. (Similar but slight more tentative conjecture, where “pullback-preserving comonad” is replaced by “pullback-preeserving modal operator”, as discussed on the page here.) I have some incomplete notes on this stuff on my web page, here, which has been left in a state of limbo. Mean to finish some thoughts here… I’ll take a look at your slides when I get a chance.

7. • CommentRowNumber7.
   • CommentAuthorMike Shulman
   • CommentTimeOct 3rd 2011
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   Author: Mike Shulman
   Format: MarkdownItexTodd: yes, you did email me about that, and replying to that email was on my to-do list as well. But now maybe I'll reply here instead, especially since the answer is simple: I don't know! The Algebraic Set Theory people have done a fair amount of work on showing that $\Pi$-W-pretoposes (sometimes with extra axioms) are stable under various constructions that toposes are known to be stable under, but I can't recall seeing lex comonads addressed anywhere, although they've considered most of the others: sheaves for internal sites, realizability, and exact completions. Do you really mean by "$\Pi$-W-pretopos" to include completeness and cocompleteness? I thought usually that term referred to the elementary notion, with only finite limits and colimits assumed.

   Todd: yes, you did email me about that, and replying to that email was on my to-do list as well. But now maybe I’ll reply here instead, especially since the answer is simple: I don’t know! The Algebraic Set Theory people have done a fair amount of work on showing that $\Pi$-W-pretoposes (sometimes with extra axioms) are stable under various constructions that toposes are known to be stable under, but I can’t recall seeing lex comonads addressed anywhere, although they’ve considered most of the others: sheaves for internal sites, realizability, and exact completions.

   Do you really mean by “$\Pi$-W-pretopos” to include completeness and cocompleteness? I thought usually that term referred to the elementary notion, with only finite limits and colimits assumed.

8. • CommentRowNumber8.
   • CommentAuthorTodd_Trimble
   • CommentTimeOct 3rd 2011
   • (edited Oct 3rd 2011)
   • PermaLink
   Author: Todd_Trimble
   Format: MarkdownItex> Do you really mean by "Π-W-pretopos" to include completeness and cocompleteness? I did mean that, but very likely idiosyncratically so! (It's just for a certain context, and I'm sure I'd be willing to drop those assumptions for a different context. Maybe I'll say complete and cocomplete $\Pi$-$W$-pretopos from now on, to avoid confusing anyone.)

   Do you really mean by “Π-W-pretopos” to include completeness and cocompleteness?

   I did mean that, but very likely idiosyncratically so! (It’s just for a certain context, and I’m sure I’d be willing to drop those assumptions for a different context. Maybe I’ll say complete and cocomplete $\Pi$-$W$-pretopos from now on, to avoid confusing anyone.)