In [this post](http://golem.ph.utexas.edu/category/2012/04/principal_bundles.html) on the *n*-Category Café, Urs Schreiber says that, "The theory of [G-principal bundles](http://ncatlab.org/nlab/show/principal+bundle) makes sense in any [$(\infty,1)$-topos](http://ncatlab.org/nlab/show/%28infinity%2C1%29-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](http://ncatlab.org/nlab/show/%28infinity%2C1%29-topos), and why is this **an** appropriate setting for the study of principal bundles, i.e., doing differential geometry?



What is an $(\infty,1)$-topos, and why is this a good setting for doing differential geometry?
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](http://people.maths.ox.ac.uk/joyce/index.html) 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.


