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I am now going through the section Structures in a cohesive oo-topos and polish and expand the discussions there.
First thing I went through is the subsection Geometric homotopy and Galois theory. It gives the definition of the fundamental -groupoid functor, a proposition on its consistency (which we had mentioned elsewhere), the definition of locally constant -stacks in the sense of -principal -bundles, and then the central theorem of Galois theory, proven by applying the -Yoneda lemma iteratively.
(This is material appearing in one form or other in other entries and at this point does not invoke the -locality, but I want to have here all in one place a nice comprehensive discussion of the whole situation in a cohesive -topos.)
I have now gone through the section Cohomology and principal oo-bundles.
I have rewritten a concise summary of the main concepts of intrinsic cohomology and then reworked the material from principal infinity-bundle, trying to make the exposition more systematic (first the definition of -principal action, then the demonstration how every homotopy fiber of a morphism is canonically equipped with such). I replaced also the material at principal infinity-bundle with this polished discussion.
at cohesive (infinity,1)-topos I have rewritten the words and expanded the content of the section Concrete objects.
(again, there is some duplication with material elsewhere on the Lab, but that is intentional: I want to collect in concise form everything relevant nicely on this page here, eventually).
I have further gone through the “Structures”-subsection:
I have parked some remarks on local over--toposes over small-projective objects and at Paths and geometric Postnikov towers (I was going to use this to discsuss -Lie algebroids over , but I need more lemmas before I can do that )
I have tried to polish the exposition at Flat -connections and local systems. I have moved a theorem and proof of what this amounts to in concrete cases to the Examples-section. Without such theorems this subsection is just a bunch of definitions, but I want all the general abstract material here separated from the implementations in the Examples-section, so i think it is good.
have also polished the exposition at de Rham cohomology and accompanied the remark that a de Rham cocycle as defined here is equivalently a flat -connection on a trivial -bundle by a (easy) theorem that makes this precise.
I have made fully detailed the previously sketchy proof of the de Rham-adjunction here.
I have added to cohesive oo-topos – de Rham cohomology a dicussion of the following:
the objects in the cohesive -topos are connected previsely if in the pieces have points - axiom holds in the sense that
is an epi.
This means that in the presence of this axiom, has essentially all the abstract properties of what is called the de Rham schematic homotopy type.
That’s maybe noteworthy.
I have added to the section Geometric homotopy and Galois theory an observation that highlights the importance of the axiom that a cohesive -topos be strongly -connected:
this implies that if one has a geometric homotopy in induced by a geometrically contractible interval object – meaning that is sufficiently connected – then under these two morphisms become equivalent.
This is important for having a large supply of geometrically contractible objects, because these identify with -Lie algebras (or rather with the objects denoted at Lie integration).
have now gone through the section Structures in a cohesive oo-topos – Differential Cohomology and polished and expanded it.
re #6:
I had defined “pieces have points” in a cohesive -topos by demanding that is an epi. That sounded a bit non-intrinsic, but did make sense.
Now it occurs to me that this is equivalent to being a regular epimorphism in an (infinity,1)-category (as now mentioned there). Which sounds much nicer.
I have now gone through the subsections
Chern-Weil homomorphisms and oo-connections and have polished and slightly expanded it;
Higher holonomy and oo-ChernSimons theory where I have replaced the discussion specific to that was there previously with a general abstract discussion, generalizing dimension of manifolds to cohomology dimension and so on. This section needs still a bit more work, but I need a break now.
okay, I have brought the last of the “Structures-” subsections into some kind of rouhgly complete state (terse at it is). I have also renamed it into
which is more accurate.
I have added to cohesive (infinity,1)-topos – structures a new section Flat Ehresmann connections which discusses how for a cohesive -group flat -Ehresmann -connections can be understood as the twisted infinity-bundles for the local coefficient -bundle
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