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The category of smooth manifolds (SmoothMfld) can be thought of the Cauchy completion of the category $U$ of open subsets of Euclidean spaces (with smooth maps) [1]. This fact is shocking to me as it provides an intrinsic definition of smooth manifolds.

If this fact can be generalized to manifolds equipped with other types of structures, we'd have a whole new perspective of what a manifold ought to be.

Alas, the proof given in [1] only works for smooth manifolds: Modulo some categoricaly nonsense, the crux of the proof (given in [1]) lies in the fact that the fixed point set of any idempotent in $U$ again has a smooth manifold structure. This essentially requires the use of tangent space and, more importantly, the inverse function theorem

Still, it doesn't prove that it fails for other cases. Thus this question: Is the category (X-Mfld) the Cauchy completion of the corresponding $U$, for X being Topological, PiecewiseLinear, Complex, Analytic.. etc?

Reference

[1] nLab authors, "chapter 4: The category of smooth manifolds", Karoubi envelope (Revision 35), August 2022.

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    $\begingroup$ I don't know the history, but I'd suspect that the fact that you mention was part of the motivation for the definition of the Cauchy completion of a category, or at least a prominent test case. That is, to speak poetically, the concept is not an elegant container into which manifolds happen coincidentally to fit, but rather an elegant container purpose-made to hold manifolds (among other things). Of course that doesn't mean it's not profitable to examine it further! $\endgroup$
    – LSpice
    Commented Aug 22, 2022 at 23:06
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    $\begingroup$ @LSpice I'm less confident about the history: I have seen Karoubi envelopes (or idempotent completions) used quite often in representation theory and category theory, where they are a very natural construction-- but I had no idea about this fact about manifolds. However, I would say this characterization of smooth manifolds seems more "extrinsic" than "intrinsic", because it really boils down to the Whitney embedding and tubular neighborhood theorems. (I.e. you are presenting the smooth manifolds by embedding them in R^n). $\endgroup$
    – Phil Tosteson
    Commented Aug 23, 2022 at 2:46
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    $\begingroup$ @PhilTosteson that's the proof, which inevitably must use extrinsic tools as we start with the extrinsic (i.e. traditional) definition. However, if we only focus on the end result, it is very intrinsic. (And yeah, such constructions show up in module theories and condensations often). $\endgroup$
    – Student
    Commented Aug 23, 2022 at 2:53
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    $\begingroup$ @LSpice That's a good point-- I think the different names for the construction might reflect different (parallel) histories of the concept. Apparently the term Cauchy completion was introduced by Lawvere. $\endgroup$
    – Phil Tosteson
    Commented Aug 23, 2022 at 3:26
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    $\begingroup$ I think the answer should be "no" for Topological, PL, and complex analytic manifolds. In the PL and topological cases, the presheaf associated to the idempotent that retracts $\mathbb R^2$ onto the coordinate axes should yield an object of the Cauchy completion which is not represented by a manifold. In the complex analytic case, compact complex analytic manifolds cannot be embedded into $\mathbb C^n$, so they should not be representable by an idempotent. $\endgroup$
    – Phil Tosteson
    Commented Aug 23, 2022 at 3:41

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(Expanding on Phil Tosteson’s comment.) No: the Cauchy-completion characterisation doesn’t hold for the PL, topological, or complex-analytic cases.

The key technical point is that split idempotents are always absolute, i.e. preserved by all functors, in particular the forgetful functor to $\mathrm{Top}$. This says that any splitting of an idempotent must be precisely (up to iso) the subspace of fixpoints, as you’d expect.

With this in hand, it’s easy to check that in the PL and topological categories, the subspace of fixpoints of an idempotent isn’t generally a manifold, so the category of manifolds isn’t Cauchy-complete. Take for instance the PL idempotent retracting $\newcommand{\R}{\mathbb{R}}\R^2$ onto the lines $x = \pm y$, sending $(x,y)$ to $\min(\left|x\right|,\left|y\right|)(\newcommand{\sg}{\operatorname{sg}}\sg(x),\sg(y))$ (where $\sg(x)$ denotes the sign of $x$, in $\{1,-1\}$).

Contrariwise, in the complex-analytic category, not every manifold arises as the splitting of an idempotent, since not every complex manifold embeds into some $\mathbb{C}^n$. For example, connected compact complex manifolds have no nonconstant holomorphic functions to $\mathbb{C}$; this follows easily from the maximum modulus principle.

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