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Has a notion of "category of Riemannian manifolds" been defined and used in the literature?

For which reasons is it or would it (not) be a useful notion?

I think the objects should be all (perhaps complete) Riemannian manifolds, and two objects should certainly be isomorphic if they are isometric as Riemannian manifolds. Which "should" be the morphisms of such a category?

I think some possibilities are:

1) isometries

2) local isometries

3) finite compositions of local isometries and Riemannian submersions

4) conformal maps

5) any of the above localized at local isometries

I tagged it "soft question" because I don't have in mind any specific application of this notion.

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    $\begingroup$ When I first saw the words "convenient category" in the title, I thought maybe you were alluding to a famous article by Norman Steenrod on a convenient category of topological spaces, where "convenient" has a technical meaning pointing mostly to the convenience of having cartesian closure for the category (for various constructions such as path spaces and the like). See ncatlab.org/nlab/show/convenient+category+of+topological+spaces. I should add that while Norman Steenrod made this famous, most of the ideas trace back to Ronnie Brown's thesis. $\endgroup$
    – Todd Trimble
    Commented Sep 1, 2012 at 13:41
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    $\begingroup$ Different categories involving Riemannian manifolds are commonly used implicitly in differential geometry, but I don't know of any significant work done where the explicit definition of a category and the use of category theory is needed. This is true in most subjects, where the main tools are analysis and PDE's, rather than algebra and topology. $\endgroup$
    – Deane Yang
    Commented Sep 1, 2012 at 14:42
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    $\begingroup$ I have thought about this sort of thing before. I wondered if you could have morphisms as Riemannian immersions and submersions. Conformal maps seem to miss too much of the structure of the objects, you might as well just talk about the category of conformal manifolds. I'd really like to see a category that captures more, rather than just the underlying metric space or smooth structure, that is also actually useful. $\endgroup$
    – Paul Reynolds
    Commented Sep 1, 2012 at 16:07
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    $\begingroup$ Could you be a little more specific as to what you mean by "should be"? You seem to imply that there is some kind of right or wrong, but you give no purpose or context for what you want to do. To me this seems like it's not well thought out even for a vague question. $\endgroup$
    – Ryan Budney
    Commented Sep 1, 2012 at 19:34
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    $\begingroup$ Agree with Ryan. It seems to me that the "right" set of objects and morphisms depends very much on what questions you're trying to answer. If you're studying conformal geometry, then obviously you want the morphisms to be at least locally conformal. If you're studying Riemannian geometry, the "right" morphisms could be global isometries, locally isometric maps, global quasi-isometries, or locally quasi-isometric maps. Or maybe something else entirely. $\endgroup$
    – Deane Yang
    Commented Sep 1, 2012 at 19:50

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I'm sure the answer to your question is "it depends on the application". Here are three categories that come to (my idiosyncratic) mind.

Perhaps the most general category in the direction you're looking is a version of Lawvere's category of metric spaces. Recall that $\mathbb R_{\geq 0}$ is a category, on account of the fact that it is a poset: there is a unique morphism $x\to y$ whenever $x\geq y$. It can be given a symmetric monoidal structure by declaring that $\otimes = +$. Then an $(\mathbb R_{\geq 0},+)$-enriched category is nothing but a (generalized) metric space. The natural "functors" are the distance-non-increasing maps.

So I would suggest that a good guess, if you must make a guess, for a category of Riemannian manifolds has as its objects all Riemannian manifolds $(M,g_M)$ (of whatever regularity you like) and its morphisms $f : (M,g_M) \to (N,g_N)$ are smooth maps $M \to N$ such that at each $m\in M$, the symmetric bilinear form $g_M - f^* g_N$ on the tangent fiber at $m$ is positive-semidefinite. This is sort of an "infinitesimal" version of the Lawvere one. Or just notice that every Riemannian manifold gives a metric space, and use the distance-nonincreasing maps (in infinite dimensions this is more subtle, as there are many important examples in which distinct points are connected by arbitrarily short paths).

But here are two other categories "of Riemannian manifolds" that are important in quantum field theory:

  1. In one approach to understanding $n$-dimensional quantum field theory (in Euclidean rather than Lorentzian signature), one constructs a category whose morphisms are $n$-dimensional compact Riemannian manifolds with boundary, and whose objects are germs of $n$-dimensional Riemannian manifolds around $(n-1)$-dimensional compact manifolds. For details on this approach, one should start with the work of Stolz and Teichner.

  2. A category I am particularly fond, which provides a "weaker" notion of $n$-dimensional Riemannian quantum field theory, of has as its objects $n$-dimensional open Riemannian manifolds, and its morphisms are isometric embeddings.

And I'm sure that there are other equally interesting categories, especially if you are interested in infinite-dimensional manifolds, which I haven't really even touched.

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