TITLE: What is an $(\infty,1)$-topos, and why is this a good setting for doing differential geometry?
QUESTION [35 upvotes]: 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?

REPLY [19 votes]: 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.