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I want to work over the site of smooth manifolds with Grothendieck topology induced by open immersion. I am primarily concerned with the the following question: in what ways can we classify vector bundles over a manifold, i.e. how can we make the functor sending manifolds to vector bundles over said manifold representable.

Classically, if $M$ is a paracompact Hausdorff space (potentially we need even less than this but I am not sure) then isomorphism classes of rank $k$ vector bundles over $M$ are in bijection with homotopy classes of maps $M\rightarrow BO_k$. In other words, while the functor from manifolds to Set given by sending a manifold to isomorphism classes of rank $k$ vector bundles over $M$ is not representable, the corresponding functor from the homotopy category of paracompact Hausdorff spaces is.

So to get a representable functor had to leave our category twice: once so that we could discuss spaces that aren't manifolds, and another time because we had "too many morphisms". I don't quite have a good feeling for why the representability problem should have a solution of this form (i.e. I understand the proof of this statement, but don't have good intuition for why this "fixed" the problem).

Alternatively, we can upgrade our vector bundle functor to instead be a pseudo functor from smooth manifolds to groupoids, where we send $M$ to the groupoid of rank $k$ vector bundles over $M$. This has no hope of being represented by a smooth manifold as the groupoids are honest groupoids and not just sets masquerading as groupoids. However, this functor is a (2,1) sheaf and so is essentially a stack by definition (this is the stack quotient of a point by $GL_k$), even more it is differentiable stack, so in the (2,1) category of differentiable stacks the object represents the functor sending a differentiable stack to it's groupoid of rank $k$ vector bundles. I think this is not so mysterious, we have by hand enlarged our category of smooth manifolds to include geometric objects which represent not representable functors in the original category.

My question is this: how are these two approaches related? I vaguely know that we can do some form of homotopy theory on the category of groupoids, and that $\pi_0$ of the vector bundle groupoid should give us the isomorphism classes, which seems to hint that there should be some connection to classical situation, but I cannot see why or how. Potentially this is just case of people using similar but ultimately different language to talk about similar but ultimately different phenomena and I have just confused myself.

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  • $\begingroup$ Can you share why you need/want representability? For example you can define a stack VBun of vector bundles over your site so that the space of bundles VBun_X over a given manifold X is the mapping space hom(X,VBun) from said manifold to VBun and in this way you can use ideas from "classifying spaces"? $\endgroup$ Commented Nov 1 at 21:13
  • $\begingroup$ @cheyne I don't need or want representability, I am just trying to make sense of how these two ideas relate to each other, and it seems to me they both are encoding some form of representability in some category. $\endgroup$ Commented Nov 1 at 21:20

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In the case under consideration, vector bundles form a sheaf of groupoids not only over the site of smooth manifolds and open maps, but also over the site of smooth manifolds and smooth maps.

For sheaves of the latter type, the smooth Oka principle provides the following very general result about such classifying spaces.

For every (∞,1)-sheaf $F$ of ∞-groupoids on the site of smooth manifolds and smooth maps, concordance classes of sections of $F$ over a smooth manifold $M$ are in a natural bijection with homotopy classes of maps from $M$ to the classifying space of $F$.

Furthermore, the latter classifying space can be explicitly computed as the homotopy colimit of the simplicial diagram, whose object of $n$-simplices is the value of $F$ on the smooth $n$-dimensional simplex.

This result admits further generalizations: it can be refined from a statement about concordance/homotopy classes to the level of spaces, and, more generally, algebras over an algebraic (∞,1)-theory. See the linked article for more information and references.

For vector bundles, $F$ is the delooping of the orthogonal Lie group (as a stack over smooth manifolds). The classifying space of $F$ is the delooping of the orthogonal ∞-group, i.e., the usual classifying space of vector bundles in topology.

Other results that follow from the smooth Oka principle include the de Rham theorem, classification of bundle gerbes, etc.

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  • $\begingroup$ I am not particularly well versed in the language of $\infinity$ categorys, but it sounds like to me you are saying that if I view this sheaf as having the highest categorical structure it can have, then $BO_k$ literally comes from this "higher stack"? $\endgroup$ Commented Nov 1 at 22:07
  • $\begingroup$ @Chris: The “highest category structure” can be taken to be the structure of a (2,1)-sheaf of groupoids, just like in the main post. As described in my answer, there is a canonical functor from such sheaves to spaces, which extracts the underlying homotopy type of a stack (alias the classifying space). Applying this functor to the (2,1)-sheaf of vector bundles indeed produces BO(k). $\endgroup$ Commented Nov 2 at 0:12
  • $\begingroup$ OK...but we still need to work in the site of smooth manifolds with smooth maps? (I am unfamiliar with this site, is there a non ncatlab source I could read about this + the smooth oka principle which you recommend?) $\endgroup$ Commented Nov 2 at 0:52
  • $\begingroup$ @Chris: Yes, but the example in the main post does come from the site of smooth manifolds. For the relation between sheaves on the etale site and the site of manifolds, see arxiv.org/abs/2309.01757 and Chapter 20 of Lurie's Spectral Algebraic Geometry. (There are functors going in each direction.) For another introduction to the smooth Oka principle, see the introduction to dmitripavlov.org/concordance.pdf. $\endgroup$ Commented Nov 2 at 1:01

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