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1. • CommentRowNumber1.
   • CommentAuthorTodd_Trimble
   • CommentTimeJul 27th 2011
   • PermaLink
   Author: Todd_Trimble
   Format: MarkdownItexIn response to a very old query at [[connected object]], I gave a proof that in an infinitary extensive category $C$, that an object $X$ is connected iff $\hom(X, -): C \to Set$ merely preserves binary coproducts. The proof was written in classical logic. If Toby would like to rework the proof so that it is constructively valid, I would be delighted.

   In response to a very old query at connected object, I gave a proof that in an infinitary extensive category , that an object is connected iff merely preserves binary coproducts.

   The proof was written in classical logic. If Toby would like to rework the proof so that it is constructively valid, I would be delighted.

2. • CommentRowNumber2.
   • CommentAuthorTobyBartels
   • CommentTimeJul 27th 2011
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   Author: TobyBartels
   Format: MarkdownItexOffhand, that proof looks irremediably nonconstructive! Now I need to think about this theorem.

   Offhand, that proof looks irremediably nonconstructive! Now I need to think about this theorem.

3. • CommentRowNumber3.
   • CommentAuthorMike Shulman
   • CommentTimeJul 29th 2011
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   Author: Mike Shulman
   Format: MarkdownItexToby: That was my reaction too. Now I'm curious whether the theorem is even true constructively.

   Toby: That was my reaction too. Now I’m curious whether the theorem is even true constructively.

4. • CommentRowNumber4.
   • CommentAuthorTodd_Trimble
   • CommentTimeJul 29th 2011
   • PermaLink
   Author: Todd_Trimble
   Format: MarkdownItexI now tend to think it's not true constructively. Any objection if I port the query box over to the Forum (or just erase it entirely)?

   I now tend to think it’s not true constructively.

   Any objection if I port the query box over to the Forum (or just erase it entirely)?

5. • CommentRowNumber5.
   • CommentAuthorDavidRoberts
   • CommentTimeJul 29th 2011
   • PermaLink
   Author: DavidRoberts
   Format: MarkdownItexI have small problem with the proof as it stands: the phrase "the canonical map $\sum_\alpha U_\alpha \to X$ is an isomorphism". This is patently false. I'm not sure what the chain of statements is supposed to be, because later we have the claim at $X \simeq U_\alpha$ for exactly one $\alpha$. Am I just being thick?

   I have small problem with the proof as it stands: the phrase “the canonical map is an isomorphism”. This is patently false. I’m not sure what the chain of statements is supposed to be, because later we have the claim at for exactly one .

   Am I just being thick?

6. • CommentRowNumber6.
   • CommentAuthorMike Shulman
   • CommentTimeJul 29th 2011
   • PermaLink
   Author: Mike Shulman
   Format: MarkdownItexWhy do you say it's false? Isn't that just part of what extensivity means? It doesn't contradict $X\cong U_\alpha$ for exactly one $\alpha$ if the other $U_\beta$ are initial.

   Why do you say it’s false? Isn’t that just part of what extensivity means? It doesn’t contradict for exactly one if the other are initial.

7. • CommentRowNumber7.
   • CommentAuthorDavidRoberts
   • CommentTimeJul 29th 2011
   • PermaLink
   Author: DavidRoberts
   Format: MarkdownItexDuh, of course. I am being thick. [hangs head in shame]

   Duh, of course. I am being thick. [hangs head in shame]

8. • CommentRowNumber8.
   • CommentAuthorTodd_Trimble
   • CommentTimeAug 3rd 2011
   • (edited Aug 3rd 2011)
   • PermaLink
   Author: Todd_Trimble
   Format: MarkdownItexSince no objections were raised, I am removing from [[connected object]] and recording here the old query box, which I hope the most recent edit has addressed: > [[Mike Shulman]]: It's not obvious to me that preserving binary coproducts is enough to ensure preservation of infinitary coproducts. Unless you meant "preserves all finite coproducts"? > _Toby_: I just copied what was at [[connected space]]. It\'s true that homming out of a connected space preserves all coproducts, but it\'s not clear to me whether that is a theorem at this level either. Maybe we need to distinguish finitarily connected objects of finitarily extensive categories from infinitarily connected objects of infinitarily extensive categories?

   Since no objections were raised, I am removing from connected object and recording here the old query box, which I hope the most recent edit has addressed:

   Mike Shulman: It’s not obvious to me that preserving binary coproducts is enough to ensure preservation of infinitary coproducts. Unless you meant “preserves all finite coproducts”?

   Toby: I just copied what was at connected space. It's true that homming out of a connected space preserves all coproducts, but it's not clear to me whether that is a theorem at this level either. Maybe we need to distinguish finitarily connected objects of finitarily extensive categories from infinitarily connected objects of infinitarily extensive categories?

9. • CommentRowNumber9.
   • CommentAuthorMike Shulman
   • CommentTimeAug 4th 2011
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   Author: Mike Shulman
   Format: MarkdownItexThanks!

   Thanks!