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    • CommentRowNumber1.
    • CommentAuthorMike Shulman
    • CommentTimeApr 5th 2014

    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?

    • CommentRowNumber2.
    • CommentAuthorTodd_Trimble
    • CommentTimeApr 5th 2014

    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.

    • CommentRowNumber3.
    • CommentAuthorMike Shulman
    • CommentTimeApr 5th 2014

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

    • CommentRowNumber4.
    • CommentAuthorMike Shulman
    • CommentTimeApr 5th 2014

    We could try MO or the catlist…

    • CommentRowNumber5.
    • CommentAuthorMike Shulman
    • CommentTimeApr 5th 2014

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

    • CommentRowNumber6.
    • CommentAuthorTodd_Trimble
    • CommentTimeApr 5th 2014
    • (edited Apr 5th 2014)

    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.

    • CommentRowNumber7.
    • CommentAuthorMike Shulman
    • CommentTimeApr 6th 2014

    Looks like you got an answer too!

    • CommentRowNumber8.
    • CommentAuthorTodd_Trimble
    • CommentTimeApr 6th 2014

    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 ε>0\epsilon \gt 0 there is always an ε\epsilon-small neighborhood VUV \subseteq U so that the original idempotent p:UUp: U \to U restricts to an idempotent on VV? 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. :-)