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There's a kind of analogy between the way manifolds work and the way bundles work. Let me try to give some examples of the analogy (although there may be better ones). I'll stick to smooth manifolds and vector bundles for definiteness.

  1. (the main idea) A manifold is $M$ a space with an open cover $\{U_i\}$ by copies of the model space $\mathbb{R}^n$, with transition functions coming from the diffeomorphism group. A vector bundle $V \to X$ is a map where the base $X$ has an open cover $\{U_i\}$ on which the map pulls back to copies of the standard map, which is a projection from $\mathbb{R}^n \times U_i \to U_i$, with transition functions required to be fiberwise in the general linear group.

  2. An orientation on a (compact) manifold $M$ is a choice of a fundamental class, i.e. $[M] \in H_n(M)$ which restricts to a generator on the local homology group $H_n(M,M-x)$ at each point $x \in M$. An orientation on a vector bundels $V \to X$ is a choice of Thom class, i.e. $v \in H^n(V,V-0)$ (where $0$ is the zero section of $V$) which restricts to a generator on the local cohomology group $H^n(V_x, V_x - 0)$ for each point $x \in X$, where $V_x$ is the fiber at $x$.

  3. Manifolds have a Pontrjagin-Thom construction, while vector bundles have a Thom construction. The idea that you can "collapse at infinity" is something in common. This connection may be me grasping at straws -- anyway, the two constructions are already closely related: the Thom construction of the normal bundle receives the map from the manifold in the Pontrjagin-Thom construction.

  4. (attempt at (1) in a more Cech-like language:) A manifold is some kind of simplicial object of model spaces. A vector bundle is some kind of simplicial object of maps to a structure group (satisfying a cocycle condition).

  5. (attempt at (1) in a more sheaf-like language:) I think a manifold $M$ is some kind of locally free sheaf on the site of Euclidean spaces and $C^\infty$-functions. The sheaf of sections of a vector bundle $V \to X$ is a locally free module over the sheaf $C^0(X)$ of continuous functions on $X$.

Point (1), in particular, leads me to want to view bundles as a "relative version of manifolds" (although vector bundles in particular happen to have a much smaller structure group). I suppose all of this could be considered in other types of geometry, too -- e.g. the same sort of analogy holds between schemes and certain kinds of sheaves over schemes. But I'd like to be able to state this in some precise way. I'd also like to understand whether the analogy stops there, or whether there's a whole hierarchy of notions of which these are just 0th-order and 1st-order notions.

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    $\begingroup$ Manifolds are spaces that are locally trivial (technically, Euclidean). Bundles are maps that are locally trivial (technically, projections from products to a factor). You forgot one more key analogy. Manifolds embed in euclidean space. Bundles have classifying maps -- meaning you can think of the manifold and all the fibres in Euclidean space. The proofs are pretty much the same. $\endgroup$ – Ryan Budney Apr 1 '17 at 4:46
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    $\begingroup$ Concerning 5. - this is almost exactly the synthetic differential geometry approach $\endgroup$ – მამუკა ჯიბლაძე Apr 1 '17 at 10:08
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Manifolds $Mfld$ and (all) G-bundles $Bndl_G$ (on arbitrary manifolds) are both (very) nice objects of the topoi associated the following respective sites:

  • Euclidean balls + $C^{\infty}$ structure.
  • The above + $G$-structure.

You should be a bit careful in the wording of your description of manifold: the chart is not part of the data! But realizing this is not pedantry, it leads to the observation that I think you want to get at: the cover gives you a presentation of a `replacement' of your manifold. (Think of this in terms of (co)fibrant replacement in model categories.)

Also, manifolds themselves, often can be constructed in a very similar fashion to presenting a vector bundle by a map $M \to BG$, where we think of this as a simplicial map from the simplicial object constructed from a cover to the simplicial version of $BG$ (rather than a stack). For example, say a manifold $M$ fibers over a base manifold $X$ into smooth fibers $N$, i.e., we have a fiber diagram $N \to M \to X$. Then $M$ is realized via pullback along a map from $X$ to the classifying stack $BDiff(N)$; and here of course one can use charts, etc., to convert to simplicial objects (which, at least for me, helps with the visualization).

Also helpful: if one considers the $\infty$-category of $\infty$-stacks (algebraic or smooth), then the simplicial versions ($\cdots G \to *$) of $BG$ coincide with its stacky versions (the functor $BG(X) =~ ${stuff on $X$}). The idea is clear: a 1-stack, in particular $BG$, is a sheaf of 1-types, and these can always be viewed as homotopy types (i.e., $\infty$-types). The simplicial version of $BG$ is just choosing a presentation.

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I think you really want to think of a vector bundle as a family of vector spaces, parametrized over a manifold and not so much like a manifold itself. I'll just point out some of the ways I see an essential distinction in your proposed analogies.

  1. It is true that vector bundle are equipped with locally trivializing patches, but these patches must respect the linear structure on the fibers of the bundle. The fundamental pieces by which smooth vector bundles (with fiber $V$) are built are projections ${\rm pr}:V\times \mathbb{R}^n\to \mathbb{R}^n$. Manifolds, by contrast, locally look like $\mathbb{R}^n$, which is an object and not a type of morphism.

  2. An orientation on a smooth manifold is equivalent to an orientation on its tangent bundle. Fix a Riemannian metric $g$ on $M$ and consider the induced map classifying the cotangent bundle of of $M$: $$g:M\to B{\rm O}(n)\;.$$ Post composing with the determinant map gives a map $M\to B\mathbb{Z}/2$, classifying the determinant line bundle $\Lambda^n(T^*M)$ over $M$. If this bundle is trivilizable, then we can choose a global nonvanishing section which is a choice of global volume form on $M$. This volume form can be used explicitly to build a Thom class for the Tangent bundle of $M$ (see for example Bott and Tu). If you look at things this way, then you see that orientability of a smooth vector bundle is really a generalization of the concept of orientability of a smooth manifold via the tangent bundle.

  3. The Pontryagin-Thom construction is basically Whitney embedding theorem + Thom construction of the normal bundle. PT and the Thom space construction are not two separate constructions that are like each other, but rather the former crucially uses the latter.

  4. The Cech and sheaf like language can be combined in a nice way in the $\infty$-topos of smooth stacks on the site of cartesian spaces (i.e. objects look like copies of $\mathbb{R}^n$). Dugger, Hollander and Isaksen show that a cofibrant replacement of a smooth manifold in Jardines model structure on simplicial presheaves is given by resolving a smooth manifold by the simplicial object given by the Cech nerve of a cover.

The smooth stack which classifies locally trivial vector bundles of rank $n$ is the homotopy orbit stack $U:=\mathbb{R}^n/\!/{\rm O}(n)$, where ${\rm O}(n)$ acts as usual, by matrix multiplication. In fact, after making some convenient choices for homotopy colimits, a map $$M\overset{\simeq}{\leftarrow} C(\{U_{\alpha}\})\to U\;,$$ where $C(\{U_{\alpha}\})$ is a homotopy colimit over the Cech nerve diagram for a cover of $M$, gives you exactly the Cech cocycles neede to define a locally trivial vector bundle on $M$. There is a canonical map $U\to \mathbf{B}{\rm O}(n)$ to the moduli stack of principal orthogonal bundles on $M$ (this map simply projects our $\mathbb{R}^n$). A map to $\mathbf{B}{\rm O}(n)$ classifies a principal bundle on $M$ and pulling back by the map $U\to \mathbf{B}{\rm O}(n)$ gives you a locally trivial vector bundle over $M$, equipped with a resolution $C(\{U_{\alpha}\times \mathbb{R}^n\})$. The sections of this object recover exactly the sheaf version of vector bundle over $M$.

In this abstract language, the relationship between manifold and vector bundle is made manifest by the fact that the local trivializing patches of a manifold prescribe the base space over which the vector bundle ought to trivialize. Still the two notions are quite distinct from one another. A vector bundle is really a map which looks like a projection over trivializing patches of $M$. The trivializing patches of $M$ are prescribed by the site of (cartesian spaces in this case).

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