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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 $\infty$-groupoid functor, a proposition on its consistency (which we had mentioned elsewhere), the definition of locally constant $\infty$-stacks in the sense of $Disc Aut(F)$-principal $\infty$-bundles, and then the central theorem of Galois theory, proven by applying the $\infty$-Yoneda lemma iteratively.
(This is material appearing in one form or other in other entries and at this point does not invoke the $\infty$-locality, but I want to have here all in one place a nice comprehensive discussion of the whole situation in a cohesive $\infty$-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 $G$-principal action, then the demonstration how every homotopy fiber of a morphism $X \to \mathbf{B}G$ 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 $n$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-$(\infty,1)$-toposes over small-projective objects $X$ and $\mathbf{\Pi}(X)$ at Paths and geometric Postnikov towers (I was going to use this to discsuss $\infty$-Lie algebroids over $X$, but I need more lemmas before I can do that )
I have tried to polish the exposition at Flat $\infty$-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 $\infty$-connection on a trivial $\infty$-bundle by a (easy) theorem that makes this precise.
I have made fully detailed the previously sketchy proof of the de Rham-adjunction $(\mathbf{\Pi}_{dR} \dashv \mathbf{\flat}_{dR})$ here.
I have added to cohesive oo-topos – de Rham cohomology a dicussion of the following:
the objects $\mathbf{\Pi}_{dR} X$ in the cohesive $\infty$-topos $\mathbf{H}$ are connected previsely if in $\mathbf{H}$ the pieces have points - axiom holds in the sense that
$\pi_0 \Gamma X \to \pi_0 \Pi X$is an epi.
This means that in the presence of this axiom, $\mathbf{\Pi}_{dR}$ 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 $\infty$-topos $\mathbf{H}$ be strongly $\infty$-connected:
this implies that if one has a geometric homotopy $f \Rightarrow g$ in $\mathbf{H}$ induced by a geometrically contractible interval object $I$ – meaning that $\Pi(I)$ is sufficiently connected – then under $\Pi : \mathbf{H} \to \infty Grpd$ these two morphisms become equivalent.
This is important for having a large supply of geometrically contractible objects, because these identify with $\infty$-Lie algebras (or rather with the objects denoted $\exp(\mathfrak{g})$ 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 $(\infty,1)$-topos by demanding that $\pi_0 \Gamma X \to \pi_0 \Pi X$ is an epi. That sounded a bit non-intrinsic, but did make sense.
Now it occurs to me that this is equivalent to $\Gamma X \to \Pi X$ 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 $\infty Lie Grpd$ 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 $G$ a cohesive $\infty$-group flat $G$-Ehresmann $\infty$-connections can be understood as the twisted infinity-bundles for the local coefficient $\infty$-bundle
$\array{ \flat_{dR} \mathbf{B}G &\to& \flat \mathbf{B}G \\ && \downarrow \\ && \mathbf{B}G } \,.$1 to 12 of 12