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In this post on the n-Category Café, Urs Schreiber says that, "The theory of G-principal bundles makes sense in any $(\infty,1)$-topos." I followed the link to the nLab and tried to chase definitions, but I found too quickly my head spinning.

What is an $(\infty,1)$-topos, and why is this an appropriate setting for the study of principal bundles, i.e., doing differential geometry?

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The idea that this is "the appropriate setting for the study of principal bundles, i.e., doing differential geometry" is ridiculous. The $(\infty, 1)$-language gives (supposedly) concrete interpretations of higher cohomology classes, which can be useful--but it's not as if the preponderance of interesting theorems in differential geometry require higher-categorical techniques to prove. –  Daniel Litt Feb 10 '13 at 7:54
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Principal bundles make sense over algebraic varieties, over topological spaces, in the category of sets, etc. There's nothing differential geometric in the idea of a principal G-bundle. Differential geometry is just one of the many worlds in which the idea of principal bundles can be interpreted. –  André Henriques Feb 10 '13 at 13:29
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It seems slightly biased to speak about any non-trivial differential geometry without differential equations, curvatures and calculus. –  Daniel Pomerleano Feb 10 '13 at 16:27
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@Daniel Litt: you have probably heard people motivate "derived algebraic geometry" as a theory refining traditional algebraic geometry? The analogous step exists for differential geometry. @Daniel Pomerleano: not sure what you are referring to, but there is differential geometry in suitable higher toposes that is all about differential equations, curvatures and calculus. @everybody: I don't know where Tom LaGatta is coming from, but I think students can easily come to think that principal bundle theory is a topic in diff. geo. for that's how many (not all) standard textbooks are written. –  Urs Schreiber Feb 10 '13 at 21:24
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@Daniel+Daniel: everybody has the liberty to focus on a small domain and not be interested on what happens elsewhere. But there are lots of people who certainly count as differential geometers and who work with orbifolds, Lie groupoids, Lie algebroids etc. This is beyond the category of smooth manifolds, but is the first step towards the oo-tops over smooth manifolds. And one is naturally led there by standard constructions, for instance the Lie integration of anything beyond finite-dimensional Lie algebras. There is loads of other motivation for which this comment box is however too small. –  Urs Schreiber Feb 11 '13 at 7:53
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4 Answers

up vote 16 down vote accepted

Derived versions of differential topology are becoming prominent tools in symplectic geometry. Whether or not you think of them via topoi is not crucial (I certainly can't), and perhaps the terminology turns off more people than it draws, but these ideas are being put to serious use by very serious no-nonsense mathematicians -- I think an excellent (though of course not isolated) example is the work of differential geometer Dominic Joyce who explains beautifully the necessities that led him deep into this area, see his 800 page book project on D-manifolds (which admittedly adapts a truncated version of the $\infty$-world for concreteness but is undoubtedly part of this story.

One way to express (very briefly) the issues is to say the derived (or $\infty$) language allows one to bypass the geometric but very subtle issues of transversality which seriously interfere with progress in some areas of geometry (Floer theory). Intersections, fiber products, and other constructions arising in moduli theory (obstructions/virtual fundamental classes) naturally lead to derived manifolds, which retain enough structure to allow algebraic constructions to work without the need for establishing and keeping track of perturbations. (This is not my area, so I can't seriously defend the need for this against a skeptic, but Joyce can..) Let me also say that this kind of geometry makes lots of geometric results (like the Atiyah-Bott fixed point theorem, some Grothendieck-Riemann-Roch and index theorems etc) completely formal. That for me is the main draw of this higher language -- it makes math that has a chance to be formal indeed formal. That's not the case for many results (and probably everything I'm saying applies more to differential topology than geometry) but that's when there are large areas where you might have dreamed that elegant abstract constructions might work but reality has proved disappointingly different, it's exciting to see that there are new languages that may (or may not) turn out up to the task.

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I am amazed by the possibility of a "purely formal" proof of GRR--is there a write-up somewhere? The Borel-Serre writeup of Grothendieck's proof is quite natural but far from formal--does the work they do move somewhere else in setting up the "derived" theory? –  Daniel Litt Feb 11 '13 at 5:19
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Do you mean by "very serious no-nonsense mathematicians" those whose dial is turned towards requiring strong motivation from more concrete problems before looking for more general, abstract settings? But then how to describe those who need a little less of a push? –  David Corfield Feb 11 '13 at 10:14
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@David - that's right, I meant mathematicians more concerned with the applications than the general structure, foundations, elegance, unity, coherence, formality... I certainly count myself among the latter and had of course no intention to slight us "nonsense mathematicians" (I realize that may have come off that way) - in particular I have great respect for the work Urs and others in this area are doing. But to other parts of the math community it's helpful to point out the certified practical need for these abstractions and potential payoff for the investment, which I think Joyce does well. –  David Ben-Zvi Feb 11 '13 at 16:57
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@David B.-Z.: I appreciate Joyce's work ever since I ran our seminar on derived diff. geom. nlab.mathforge.org/schreiber/show/… . Back then he used to announce a major application in the theory of certain moduli stacks. Has that meanwhile been published? While all this is great, I was a bit surprised about your reply here. I thought the question was about higher toposes and principal bundles and differential geometry. If the question is more generally: what is derived/higher geometry, loads of people besides Joyce come to mind who contributed. –  Urs Schreiber Feb 11 '13 at 17:35
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@Urs you're absolutely right, I ignored the main two-thirds of the question (what is a higher topos and what are principal bundles on it) and focused on the (implied?) question about advantages of doing differential geometry in some sort of derived/higher setting, which the comments seemed to address.. Certainly Joyce is one of many people working in the field, but his research statements/surveys on his webpage provide IMHO a strong case for the concrete need for these ideas. –  David Ben-Zvi Feb 11 '13 at 18:05
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Ok, so, I will try to answer this best I can. first, I'll tell you a skewed-perspective of what an infinity topos is (or ought-to-be). As for how you can "do differential geometry"- this is a bold statement. But, there's certain aspects of differential geometry which extend naturally / have a nice interpretation in (certain) infinity topoi (with extra structure).

What is an infinity topos? It certainly can wear many different cloaks, and I won't attempt to give a global overview; there's reason that even solid books on $1$-topoi are quite long, since topoi can be thought of as generalized spaces, or theories in logic, or universes that behave like Set, etc.. I will concentrate on just one particular aspect of infinity topos theory. You may have heard the slogan "a topos is a category that behaves like the category of sets". In this vain, the analogous slogan is "an infinity topos is an infinity category that behaves like the infinity category of spaces (thought of as homotopy types, i.e. infinity groupoids)." Basically, an infinity topos provides one with a place in which to do homotopy theory. Objects in an infinity topos have homotopy groups, you can talk about Eilenberg-Mac Lane objects etc. To be more concrete, just as a 1-topos arises by taking sheaves of sets on a 1-category, an infinity topos arises as taking "sheaves" of spaces (infinity groupoids) over an infinity category. (Although, the concept of sheaf is different AND to be more correct, every infinity topos arises as a so-called cotopological localization of an infinity category of infinity sheaves on an infinity category- this is due to the failure of Whitehead's theorem internal to the sheaf topos) Basically, one should think of an infinity topos of sheaves (or some hypercompletion thereof etc.) on a category $C$ to be some sort of hybredization of "generalized objects of $C$" and infinity groupoids. What do I mean? Well, sometimes people view (at least certain) sheaves on the category of manifolds as generalized manifolds. In fact, you can faithfully represent all infinite dimensional manifolds this way. What is an example of something between a manifold and a groupoid? An orbifold. Orbifolds (and more general differentiable stacks) naturally lives in the 2-topos of sheaves (stacks) of groupoids on manifolds. An orbifold / differentialble stack is like a manifold whose points can posses finite / Lie intrinsic symmetry groups. What if these symmetry groups themselves didn't form a manifold, but another differentiable stack? Then you would be looking at a higher differentiable stack (this one would live in the 3-topos). In general, everything lives in the infinity topos of sheaves of infinity groupoids on manifolds.

Ok, what does this have to do with principal bundles? If you're given a Lie group $G,$ one can consider this as a group object in manifolds, hence a groupoid object in manifolds (with one object), and hence a (representable) sheaf of groupoids on manifolds. It's not a stack though, but the stack it represents, is the functor $$Mfd^{op} \to Gpd$$ sending a manifold $M$ to the groupoid of principal $G$-bundles over $M$. This stack, is often denoted by $BG$ and is quite formally related to the topological classifying space (and in fact, as a differentiable stack, has the same homotopy type as this). By the Yoneda lemma, maps (not homotopy classes, ALL maps) from $M$ to $BG$ are the same as principal $G$-bundles on $M,$ and $BG$ caries a universal principal $G$-bundle over itself, just as in the topological picture. Now, suppose you cared about line bundles, then you could let $G=U(1)$ and then $BU(1).$ Suppose instead, you cared about bundle-gerbes, then you take $B^2U(1),$ which corresponds to viewing $U(1)$ as a $2$-groupoid with one object, and for bundle $2$-gerbes $B^3U(1),$ etc.

Ok, but how is this differential geometry? Right, so, it isn't. Not yet. I don't think it's accurate to say you can "do differential geometry" in an infinity topos. What is true though, is that there are many interesting infinity topoi with extra structure which in addition to having a notion of principal bundle etc., have a good notion of principal bundle with connection and allow you make good sense of "differential cohomology". (You can also make sense of things like "higher" Lie theory etc.). What Urs noticed is, you can define all these things inside any infinity topos admitting a "cohesive structure". I don't expect the definition to be enlightening. The point is, inside any "cohesive infinity topos" one can make sense of principal bundles with connection, and more generally, basically everything you need to make sense of so-called "higher gauge theory" (Urs' motivation comes from physics). E.g., the language lets you make good sense of what is a smooth $String(n)$-bundle with connection- or what a smooth $FiveBrane(n)$-bundle is, with connection. This is what Urs means when he says you can "do differential geometry."

Anyhow, I hope Urs himself chimes in as well.

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Here is one way to say it, which makes the relation to principal bundle theory most manifest ( http://ncatlab.org/schreiber/show/Principal+%E2%88%9E-bundles+--+theory,+presentations+and+applications ):

An $\infty$-topos is a context for homotopy theory that satisfies three extra axioms, the "Giraud-Rezk-Lurie"-axioms (for all keywords see the pointers behind the above link). In

Thomas Nikolaus, Urs Schreiber, Danny Stevenson Principal infinity-bundles - General theory http://arxiv.org/abs/1207.0248

it is shown that precisely two of these axioms make the theory of principal bundles work well, with their classification by nonabelian cohomology. This is purely axiomatic, hence completely general. It can for instance be implemented in homotopy type theory.

In the second part

Thomas Nikolaus, Urs Schreiber, Danny Stevenson Principal infinity-bundles - Presentations http://arxiv.org/abs/1207.0249

are discussed convenient ways to implement this general theory in geometric contexts such as topology, differential geometry, algebraic geometry, etc. This way the axiomatic theory recovers tradional theory, inclduding theory of gerbes, higher gerbes, twisted cohomology, twisted bundles, simplicial bundles etc.

There is announced a thrid part "Principal infinity-bundles - Examples and applications", which is not out yet. But loads of examples and applications are discussed in the text

Differential cohomology in a cohesive topos http://ncatlab.org/schreiber/show/differential+cohomology+in+a+cohesive+topos

There are corresponding lecture notes

The geometry of physics http://ncatlab.org/nlab/show/geometry+of+physics

A few weeks back in January I was giving the first three lectures of the second part of these notes in Singapore at the TFT meeting. If you have any familiary with physics, check out for instance the session

geometry fof physics - Fields http://ncatlab.org/nlab/show/geometry%20of%20physics#Fields

which includes a discussion of how the traditional notion of "field bundles" in quantum field theory is not completely correct, and how its correction leads one to studying bundle theory in an infinity-topos. Lots of examples follow.

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Thank you so much for your comprehensive references in both answers, Urs. I accepted David Ben-Zvi's answer since it provides the motivation to studying differential geometry from the general perspective, and your list of resources provide an invaluable followup. I look forward to understanding more of your point of view. –  Tom LaGatta Feb 11 '13 at 20:04
    
Had you asked for motivation, I would have provided plenty. Check out this commented list here: ncatlab.org/nlab/show/motivation+for+higher+differential+geometry –  Urs Schreiber Feb 12 '13 at 20:45
    
and now with hyperlink: ncatlab.org/nlab/show/… –  Urs Schreiber Feb 12 '13 at 20:54
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If the question is not really about principal bundle theory but just about: why do we need higher differential geometry at all, then of course there are plenty of further answers:

Classical differential geometry includes orbifolds

http://ncatlab.org/nlab/show/orbifold

as objects that handle non-free quotients of smooth manifolds. These are really the first kinds of examples of Lie groupoids

http://ncatlab.org/nlab/show/Lie+groupoid

hence of stacks on the category of smooth manifolds. All of classical foliation theory

http://ncatlab.org/nlab/show/foliation

is secretly Lie groupoid theory. And the only sane way of understanding the collection of Lie groupoids, with their correct notion of Morita equivalence and of homotopy, is as understanding them as the objects of the $(2,1)$-topos of stacks over smooth manifolds.

Also Lie theory itself breaks out of the category of smooth manifolds. For instance where Lie's three theorems fail: not every infinite-dimensional Lie algebra integrates to a Lie group, but it instead integrates to a certain Lie 2-group, a group object in Lie groupoids/smooth stacks. This is all the more true as soon as you admit that Lie algebroids are part of differential geometry

http://ncatlab.org/nlab/show/Lie+algebroid .

Lie algebroids directly encode PDEs

http://ncatlab.org/nlab/show/exterior+differential+system

Most Lie algebroids integrate to Lie groupoids, but some want to integrate to Lie 2-groupoids, which liven in the $(3,1)$-topos over smooth manifolds. And so on.

Doing differential geometry and not stopping when classical constructions fail invariably leads one to higher differential geometry, hence to working in the $(\infty,1)$-topos over smooth manifolds.

Maybe that's what the question was really asking. If in addition one feels like refining the site of smooth manifolds itself such as to include "derived smooth manifolds", then one gets something even richer, as mentioned in another reply here. Such derived and higher differential geometry is notably the home of BV-BRST formalism

http://ncatlab.org/nlab/show/BV-BRST+formalism

hence of the modern form of variational calculus, symplectic reduction and homological integration theory.

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