Any space XX can always be turned into a manifold modelled on itself, using any pseudogroup GG. Simply take the inclusions of open sets as charts.

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Bernhard Riemann, Über die Hypothesen, welche der Geometrie zu Grunde liegen, Göttingen (1845) [doi:10.1007/978-3-642-35121-1]

We will present two possible definitions. The first, via pseudogroups, has a simpler definition, but has two (rather serious) drawbacks:

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If M 1M_1 and M 2M_2 are two GG-manifold structures on the same topological space MM, then M 1M_1 and M 2M_2 are considered equal as GG-manifolds if id:M→Mid: M \to M is an isomorphism from M 1M_1 to M 2M_2 (and hence also from M 2M_2 to M 1M_1).

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A manifold is a topological space which looks locally like a Cartesian space, commonly a finite-dimensional Cartesian space ℝ n\mathbb{R}^n, in which case one speaks of a manifold of dimension nn or nn-fold, but possibly an infinite-dimensional topological vector space, in which case one has an infinite-dimensional manifold.

where ff is continuous and ii is an open embedding. The spans are locally (that is, for fixed XX and YY) ordered by inclusion.

A topological nn-manifold is a manifold with respect to the pseudogroup of homeomorphisms between open sets of ℝ n\mathbb{R}^n. Any continuous function between topological manifolds is a morphisms, and any homeomorphism is an isomorphism. A piecewise-linear (PL) n n -manifold is where the pseudogroup consists of piecewise-linear homeomorphisms between such open sets; morphisms are called piecewise-linear (PL) maps.

Toby: Yes, I think that this would make perfect sense; I think that we'd want GG to be an internal groupoid in CC. Note that defining things like ‘smooth manifold’ in CC might still be difficult, but we've reduced it to internalising Cart Sp in CC. (There's also the matter that the above definition takes a notion of space for granted, so you'd have to internalise that into CC too, but I'm not sure how important that really is, when I think about how the topology on a smooth manifold can be recovered from the smooth structure.)

We begin by defining the 2-poset (i.e., locally preordered bicategory) of regions, denoted RegReg. The objects are topological spaces (or locales if you prefer); the morphisms are partial functions with open domain, that is spans

If the term “manifold” appears without further qualification, what is usually meant is a smooth nn-manifold of some natural number dimension nn: a GG-manifold where GG is the pseudogroup of invertible C ∞C^{\infty} maps between open sets of ℝ n\mathbb{R}^n. Replacing ℝ n\mathbb{R}^n here by a half-space {x∈ℝ n:x 1≥0}\{x \in \mathbb{R}^n: x_1 \geq 0\}, one obtains the notion of smooth manifold with boundary. Or, replacing ℝ n\mathbb{R}^n here by the nn-cube I nI^n, one obtains the notion of (smooth) nn-manifold with (cubical) corners. Morphisms of manifolds are here called smooth maps, and isomorphisms are called diffeomorphisms. (In manifold theory, one usually reserves the term smooth function for smooth maps to ℝ\mathbb{R}.)

What “locally looks like” means depends on what sort of structure we are considering a Cartesian space to embody. At one extreme, we can think of ℝ n\mathbb{R}^n as merely a topological space. Or, ℝ n\mathbb{R}^n may be considered as carrying more rigid types of structure, such as C kC^k-differential structure, smooth structure, piecewise-linear (PL) structure, real analytic structure, affine structure, hyperbolic structure, foliated structure, etc., etc. Accordingly we have notions of topological manifold, differentiable manifold, smooth manifold, analytic manifold etc. By default these are modeled on finite dimensional spaces, but most notions have generalizations to a corresponding notion of infinite dimensional manifold.

In many of the typical cases, this will mean that MM is metrizable. In many studies, for example in cobordism theory, one goes even further and assumes the manifolds are compact.

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Toby: I'd rather not, since it's all ‘I think’ and ‘might be difficult’; it's better as a query box, moved to the bottom if necessary. But if Todd agrees with me, then maybe he'll add it.

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Here we will focus on the general notion of a manifold. More concrete examples can be found in individual pages such as topological manifold and smooth manifold.

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Given a cartology CC, a morphism r=(U,f):X→Yr = (U, f): X \to Y in CC is pseudo-invertible if there exists s=(V,g):Y→Xs = (V, g): Y \to X such that s∘r=1 Us \circ r = 1_U and r∘s=1 Vr \circ s = 1_V.

is in GG whenever (U,ϕ)(U, \phi) is a coordinate chart, def. of x∈Mx \in M, and (V,ψ)(V, \psi) is a coordinate chart of f(x)∈Nf(x) \in N.

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satisfying the expected properties. We can thus speak of CC-manifolds (or (C,X)(C, X)-manifolds if we want to make explicit the modeling space XX).

Engl. transl: William Clifford: On the hypotheses which underlie geometry, Nature VIII (1873) 183-184 [doi:10.1007/978-3-319-26042-6]

A GG-atlas on a topological space MM is a family of GG-compatible charts (ϕ α:U α→X) α(\phi_\alpha: U_\alpha \to X)_\alpha, def. , such that (U α) α(U_\alpha)_\alpha covers MM. The (restricted) maps ϕ αβ=ϕ β∘ϕ α −1\phi_{\alpha \beta} = \phi_\beta \circ \phi_{\alpha}^{-1} are called transition functions between the charts of the atlas.

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An isomorphism of GG-manifolds f:M→Nf: M \to N (defined by chosen atlas structures, def. ) is a homeomorphism ff such that

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These definitions need to be carefully checked against known examples (e.g., the categories TopTop, PLPL, and SmoothSmooth, among others).

An atlas is not considered an essential part of the structure of a manifold: two different atlases may yield the same manifold structure. This is encoded by the following definition of isomorphisms between manifolds.

whose opposite i op:X→Ext(r)i^op: X \to Ext(r) also belongs to RegReg (that is, ii is an open embedding), and the equations

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The setting is a topological space XX together with a pseudogroup GG on XX. For the sake of concreteness, the reader may as well focus on the case X=ℝ nX = \mathbb{R}^n and GG is the groupoid of diffeomorphisms between open subsets of XX.

Note: the following is tentative “original research”. It is prompted by the desire to extend the pseudogroup approach for defining general notions of manifold, so as to cover also an appropriate general notion of “map”. Comments, improvements, and corrections are encouraged – Todd.

It is almost invariably the case in classical manifold theory that one requires some technical niceness properties on the topological space underlying a manifold.

Rafael: Can one define a manifold object in a category C as a G-manifold with G related to C? What would the relation between G and C be to obtain G-manifolds in C as manifold objects?

Now, given a cartology CC, we define the category of CC-manifolds. Let MM be a (C,X)(C, X)-manifold and NN a (C,Y)(C, Y)-manifold. Then, a CC-morphism from MM to NN is a continuous map f:M→Nf: M \to N such that the RegReg-composite

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The notion of a CC-manifold modeled on an object XX of CC is defined just as before, using the pseudogroup on XX implied by the previous lemma. In particular, we have CC-charts of an atlas structure on MM, which are morphisms in RegReg

hold. The object Ext(r)Ext(r) may be called the extension of rr. This splitting is a kind of comprehension principle? familiar from the theory of allegories, among other things.

Alternatively, atlases are ordered by inclusion, and two atlases define the same manifold structure on MM if they have a common upper bound. Equivalently, two atlases define the same manifold structure if each chart of one is compatible with each chart of the other. Or, one could extend any atlas to the (unique) maximal atlas containing it, which consists of all charts compatible with each of the charts in the original atlas, and simply identify a manifold structure with a maximal atlas.

Rafael: Can someone that knows more than me about this add the result of this question to this article so nobody have to ask again.

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One can go on to define, in a straightforward way, real analytic manifolds, complex analytic manifolds, elliptic manifolds, hyperbolic manifolds, and so on, using the general notion of pseudogroup.

In any case, the type of geometry embodied in a particular flavor of manifold is controlled by a particular groupoid or, more generally, category of transformations which preserves whatever geometric features one is interested in; cf. Felix Klein’s Erlanger Programm.