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1. • CommentRowNumber1.
   • CommentAuthorMike Shulman
   • CommentTimeApr 5th 2014
   • PermaLink
   Author: Mike Shulman
   Format: MarkdownItexDo we have recorded on the nLab anywhere facts about what limits exist in the category of (ordinary, finite-dimensional) manifolds? Pullbacks of submersions, right? What equalizers?

   Do we have recorded on the nLab anywhere facts about what limits exist in the category of (ordinary, finite-dimensional) manifolds? Pullbacks of submersions, right? What equalizers?

2. • CommentRowNumber2.
   • CommentAuthorTodd_Trimble
   • CommentTimeApr 5th 2014
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   Author: Todd_Trimble
   Format: MarkdownItexWell, I would _love_ to firm up the claim (made at [[Karoubi envelope]] I think, and maybe elsewhere in the Lab) that the category of smooth manifolds is Cauchy-complete. This is something Lawvere said once, and I certainly believe it, but I've made concerted efforts to find this in the literature and have come up empty. And I never did cook up a detailed proof myself; if anyone can provide that, I'd really love to see it. It's not hard to put stuff together from the literature that implies that every smooth manifold is a smooth retract of an open space of Euclidean space. So the category of smooth manifolds sits inside the Cauchy completion of open sets and smooth maps. But I was having a devil of a time trying to convince myself this embedding is essentially surjective.

   Well, I would love to firm up the claim (made at Karoubi envelope I think, and maybe elsewhere in the Lab) that the category of smooth manifolds is Cauchy-complete. This is something Lawvere said once, and I certainly believe it, but I’ve made concerted efforts to find this in the literature and have come up empty. And I never did cook up a detailed proof myself; if anyone can provide that, I’d really love to see it.

   It’s not hard to put stuff together from the literature that implies that every smooth manifold is a smooth retract of an open space of Euclidean space. So the category of smooth manifolds sits inside the Cauchy completion of open sets and smooth maps. But I was having a devil of a time trying to convince myself this embedding is essentially surjective.

3. • CommentRowNumber3.
   • CommentAuthorMike Shulman
   • CommentTimeApr 5th 2014
   • PermaLink
   Author: Mike Shulman
   Format: MarkdownItexYou can't get something like the cross $\{x=0\}\cup\{y=0\}$ as a smooth retract of $\mathbb{R}^2$?

   You can’t get something like the cross $\{x=0\}\cup\{y=0\}$ as a smooth retract of $\mathbb{R}^2$?

4. • CommentRowNumber4.
   • CommentAuthorMike Shulman
   • CommentTimeApr 5th 2014
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   Author: Mike Shulman
   Format: MarkdownItexWe could try MO or the catlist...

   We could try MO or the catlist…

5. • CommentRowNumber5.
   • CommentAuthorMike Shulman
   • CommentTimeApr 5th 2014
   • PermaLink
   Author: Mike Shulman
   Format: MarkdownItexIt's unfortunately difficult to search google for "equalizer of manifolds"...

   It’s unfortunately difficult to search google for “equalizer of manifolds”…

6. • CommentRowNumber6.
   • CommentAuthorTodd_Trimble
   • CommentTimeApr 5th 2014
   • (edited Apr 5th 2014)
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   Author: Todd_Trimble
   Format: MarkdownItexYeah, I had tried "fixed point set idempotent smooth manifold" and other things, and found lots of interesting material, but no cigar. Anyway, I went ahead and [posted](http://mathoverflow.net/questions/162552/idempotents-split-in-category-of-smooth-manifolds) my question to MO.

   Yeah, I had tried “fixed point set idempotent smooth manifold” and other things, and found lots of interesting material, but no cigar. Anyway, I went ahead and posted my question to MO.

7. • CommentRowNumber7.
   • CommentAuthorMike Shulman
   • CommentTimeApr 6th 2014
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   Author: Mike Shulman
   Format: MarkdownItexLooks like you got an answer too!

   Looks like you got an answer too!

8. • CommentRowNumber8.
   • CommentAuthorTodd_Trimble
   • CommentTimeApr 6th 2014
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   Author: Todd_Trimble
   Format: MarkdownItexYes, I did. There was a small point that I decided not to bother the answerer with (how do we a priori know that for any $\epsilon \gt 0$ there is always an $\epsilon$-small neighborhood $V \subseteq U$ so that the original idempotent $p: U \to U$ restricts to an idempotent on $V$? that's the starting part of his argument). The argument I wrote up at [Karoubi envelope](http://ncatlab.org/nlab/show/Karoubi+envelope#the_category_of_smooth_manifolds) doesn't quite match the brevity of his argument, but it does take care of the small point that was bothering me. :-)

   Yes, I did. There was a small point that I decided not to bother the answerer with (how do we a priori know that for any $\epsilon \gt 0$ there is always an $\epsilon$-small neighborhood $V \subseteq U$ so that the original idempotent $p: U \to U$ restricts to an idempotent on $V$? that’s the starting part of his argument). The argument I wrote up at Karoubi envelope doesn’t quite match the brevity of his argument, but it does take care of the small point that was bothering me. :-)