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1. • CommentRowNumber1.
   • CommentAuthorDavidRoberts
   • CommentTimeMay 8th 2013
   • (edited May 9th 2013)
   • PermaLink
   Author: DavidRoberts
   Format: MarkdownItexCreated the page [[unbounded topos]], and some links at [[topos]] and [[bounded geometric morphism]]. I'm interested in the generalisation of the construction of the unbounded topos $Gl(F)$ to the general case of an inaccessible comonad $G$ on a bounded topos (which wlog we might as well take to be $Set$ EDIT: NO, LET'S NOT). In essence, _why_ is it unbounded? Also, what nice properties can we claim of the category of coalgebras for $G$, given information about $G$. Note also, the paper HOW LARGE ARE LEFT EXACT FUNCTORS? in TAC in 2001 seems to claim something a little stronger than Johnstone does in the Elephant, and recounted at [[topos]], namely that the existence of lex endofunctors of set is independent of ZFC (they say something more general, but it covers this case). This is mostly a note to myself, but if others feel like looking, that would be good too.

   Created the page unbounded topos, and some links at topos and bounded geometric morphism.

   I’m interested in the generalisation of the construction of the unbounded topos $Gl(F)$ to the general case of an inaccessible comonad $G$ on a bounded topos (which wlog we might as well take to be $Set$ EDIT: NO, LET’S NOT). In essence, why is it unbounded? Also, what nice properties can we claim of the category of coalgebras for $G$, given information about $G$.

   Note also, the paper HOW LARGE ARE LEFT EXACT FUNCTORS? in TAC in 2001 seems to claim something a little stronger than Johnstone does in the Elephant, and recounted at topos, namely that the existence of lex endofunctors of set is independent of ZFC (they say something more general, but it covers this case). This is mostly a note to myself, but if others feel like looking, that would be good too.

2. • CommentRowNumber2.
   • CommentAuthorMike Shulman
   • CommentTimeMay 8th 2013
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   Author: Mike Shulman
   Format: MarkdownItex> which wlog we might as well take to be Set Hmm, I'm not sure about that. What if the inaccessible comonad $G$ is not an indexed comonad, so that it's not visible to the internal logic of the topos in question? > seems to claim something a little stronger It does!

   which wlog we might as well take to be Set

   Hmm, I’m not sure about that. What if the inaccessible comonad $G$ is not an indexed comonad, so that it’s not visible to the internal logic of the topos in question?

   seems to claim something a little stronger

   It does!

3. • CommentRowNumber3.
   • CommentAuthorDavidRoberts
   • CommentTimeMay 9th 2013
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   Author: DavidRoberts
   Format: MarkdownItex>Hmm, I’m not sure about that. Actually, that's precisely _not_ what I wanted to assume! So I take that back.

   Hmm, I’m not sure about that.

   Actually, that’s precisely not what I wanted to assume! So I take that back.

4. • CommentRowNumber4.
   • CommentAuthorTobyBartels
   • CommentTimeMay 19th 2013
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   Author: TobyBartels
   Format: MarkdownItexIs it possible to have an unbounded geometric morphism from a Grothendieck topos to $Set$? (Necessarily, this would be different from the global-points functor, which is bounded.) The definition spoke of the topos itself as being unbounded, but it\'s really the topos equipped with the geometric morphism that we\'re talking about, so I changed this to clarify that. If every geometric morphism from a Grothendieck topos to $Set$ must be bounded, then it doesn\'t really matter. But a Grothendieck topos could be an example of an unbounded topos if it were equipped with an unbounded geometric morphism to $Set$ rather than with the global-points functor.

   Is it possible to have an unbounded geometric morphism from a Grothendieck topos to $Set$? (Necessarily, this would be different from the global-points functor, which is bounded.) The definition spoke of the topos itself as being unbounded, but it's really the topos equipped with the geometric morphism that we're talking about, so I changed this to clarify that. If every geometric morphism from a Grothendieck topos to $Set$ must be bounded, then it doesn't really matter. But a Grothendieck topos could be an example of an unbounded topos if it were equipped with an unbounded geometric morphism to $Set$ rather than with the global-points functor.

5. • CommentRowNumber5.
   • CommentAuthorDavidRoberts
   • CommentTimeMay 19th 2013
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   Author: DavidRoberts
   Format: MarkdownItexIsn't there at most, up to isomorphism, one geometric morphism to Set from any topos?

   Isn’t there at most, up to isomorphism, one geometric morphism to Set from any topos?

6. • CommentRowNumber6.
   • CommentAuthorTobyBartels
   • CommentTimeMay 19th 2013
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   Author: TobyBartels
   Format: MarkdownItexOK, that would settle it then!

   OK, that would settle it then!