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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 $C$, 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.

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
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   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 $\sum_\alpha U_\alpha \to X$ 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?

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 $X\cong U_\alpha$ for exactly one $\alpha$ if the other $U_\beta$ are initial.

7. • CommentRowNumber7.
   • CommentAuthorDavidRoberts
   • CommentTimeJul 29th 2011
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   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!