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](https://ncatlab.org/nlab/revision/Karoubi%20envelope/35#the_category_of_smooth_manifolds)", [*Karoubi envelope* (Revision 35)](http://ncatlab.org/nlab/revision/Karoubi%20envelope/35), August 2022.



Manifolds as Cauchy completed objects
(Expanding on Phil Tosteson’s [comment](https://mathoverflow.net/questions/429003/manifolds-as-cauchy-completed-objects#comment1103552_429003).) **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.


