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1. • CommentRowNumber1.
   • CommentAuthorTodd_Trimble
   • CommentTimeSep 19th 2011
   • PermaLink
   Author: Todd_Trimble
   Format: MarkdownItexI added a comment to the end of the discussion at [[predicative mathematics]] to the effect that free small-colimit completions of toposes are examples of locally cartesian closed pretoposes that are generally not toposes.

   I added a comment to the end of the discussion at predicative mathematics to the effect that free small-colimit completions of toposes are examples of locally cartesian closed pretoposes that are generally not toposes.

2. • CommentRowNumber2.
   • CommentAuthorTodd_Trimble
   • CommentTimeSep 19th 2011
   • (edited Sep 19th 2011)
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   Author: Todd_Trimble
   Format: MarkdownItexIn fact, there could be some interesting stability theorems here. For one, if $C$ is a complete and cocomplete $\Pi$-pretopos, then we have an evident functor $\Delta: Set \to C$ and I believe * $C \downarrow \Delta$ is also a complete and cocomplete $\Pi$-pretopos. (Note: it may be shown that $C \downarrow \Delta$, for $C$ an $\infty$-lextensive category, is the free small-coproduct completion of $C$.) Furthermore, if $E_{ex}$ denotes the ex/lex completion, then * $E_{ex}$ is a complete and cocomplete $\Pi$-pretopos if $E$ is. One import of this is that the free small-colimit completion $Colim(C)$, for $C$ a complete and cocomplete $\Pi$-pretopos, is equivalent to $(C \downarrow \Delta)_{ex}$, which by the above is also a complete and cocomplete $\Pi$-pretopos. Furthermore, I conjecture that the above stability theorems carry over to $\Pi$-$W$-pretoposes in place of $\Pi$-pretoposes. These constructions seem to give interesting ways of constructing lots of non-topos examples.

   In fact, there could be some interesting stability theorems here. For one, if $C$ is a complete and cocomplete $\Pi$-pretopos, then we have an evident functor $\Delta: Set \to C$ and I believe

   • $C \downarrow \Delta$ is also a complete and cocomplete $\Pi$-pretopos.

   (Note: it may be shown that $C \downarrow \Delta$, for $C$ an $\infty$-lextensive category, is the free small-coproduct completion of $C$.) Furthermore, if $E_{ex}$ denotes the ex/lex completion, then

   • $E_{ex}$ is a complete and cocomplete $\Pi$-pretopos if $E$ is.

   One import of this is that the free small-colimit completion $Colim(C)$, for $C$ a complete and cocomplete $\Pi$-pretopos, is equivalent to $(C \downarrow \Delta)_{ex}$, which by the above is also a complete and cocomplete $\Pi$-pretopos.

   Furthermore, I conjecture that the above stability theorems carry over to $\Pi$-$W$-pretoposes in place of $\Pi$-pretoposes. These constructions seem to give interesting ways of constructing lots of non-topos examples.

3. • CommentRowNumber3.
   • CommentAuthorTobyBartels
   • CommentTimeSep 19th 2011
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   Author: TobyBartels
   Format: MarkdownItexCool!

   Cool!

4. • CommentRowNumber4.
   • CommentAuthorTodd_Trimble
   • CommentTimeSep 20th 2011
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   Author: Todd_Trimble
   Format: MarkdownItexIf you have a left exact (or even pullback-preserving) comonad on a pretopos, is the category of coalgebras also a pretopos? Same question for $\Pi$-pretoposes.

   If you have a left exact (or even pullback-preserving) comonad on a pretopos, is the category of coalgebras also a pretopos? Same question for $\Pi$-pretoposes.

5. • CommentRowNumber5.
   • CommentAuthorTodd_Trimble
   • CommentTimeSep 21st 2011
   • (edited Sep 21st 2011)
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   Author: Todd_Trimble
   Format: MarkdownItexMy question in 4 is in fact an incredibly easy 'yes' (on both counts). Well, certainly the pretopos part is: all the axioms of a pretopos are in terms of colimits and pullbacks. If $G: E \to E$ is a pullback-preserving comonad, then the underlying functor $U: Coalg_G \to E$ preserves and reflects colimits and pullbacks, and this means that the pretopos axioms hold on $Coalg_G$ if they hold on $E$. Nothing could be simpler. (It makes me suspect that a very easy conceptual proof of the lex comonad theorem for toposes might be available if we 'define' a topos to be a lex total category, which is not too far off from the truth for Grothendieck toposes.) The analogous result for $\Pi$-pretoposes is also pretty easy if you quote the result (say from the Elephant) that if $E$ is locally cartesian closed, then so is $Coalg_G$. (I am defining a $\Pi$-pretopos to be a cocomplete LCC pretopos; it is easy to see that a $\Pi$-pretopos is also complete.) I have been writing all this down on my web page <a href="http://ncatlab.org/toddtrimble/published/Small+cocompletions">here</a>, including proofs of my assertions in 2 above (I haven't tackled $W$-types yet). Some of it might be a bit clunky. But the basic upshot is that the free small-colimit cocompletion of a $\Pi$-pretopos is also a $\Pi$-pretopos (and almost never a topos). It bothers me though that all this is a bit abstract; I'm having trouble *picturing* the nature of these constructions.

   My question in 4 is in fact an incredibly easy ’yes’ (on both counts). Well, certainly the pretopos part is: all the axioms of a pretopos are in terms of colimits and pullbacks. If $G: E \to E$ is a pullback-preserving comonad, then the underlying functor $U: Coalg_G \to E$ preserves and reflects colimits and pullbacks, and this means that the pretopos axioms hold on $Coalg_G$ if they hold on $E$. Nothing could be simpler.

   (It makes me suspect that a very easy conceptual proof of the lex comonad theorem for toposes might be available if we ’define’ a topos to be a lex total category, which is not too far off from the truth for Grothendieck toposes.)

   The analogous result for $\Pi$-pretoposes is also pretty easy if you quote the result (say from the Elephant) that if $E$ is locally cartesian closed, then so is $Coalg_G$. (I am defining a $\Pi$-pretopos to be a cocomplete LCC pretopos; it is easy to see that a $\Pi$-pretopos is also complete.)

   I have been writing all this down on my web page here, including proofs of my assertions in 2 above (I haven’t tackled $W$-types yet). Some of it might be a bit clunky. But the basic upshot is that the free small-colimit cocompletion of a $\Pi$-pretopos is also a $\Pi$-pretopos (and almost never a topos). It bothers me though that all this is a bit abstract; I’m having trouble picturing the nature of these constructions.