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    • CommentRowNumber1.
    • CommentAuthorUrs
    • CommentTimeJan 7th 2011
    • (edited Jan 7th 2011)

    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 DiscAut(F)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.)

    • CommentRowNumber2.
    • CommentAuthorUrs
    • CommentTimeJan 7th 2011
    • (edited Jan 7th 2011)

    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 GG-principal action, then the demonstration how every homotopy fiber of a morphism XBGX \to \mathbf{B}G is canonically equipped with such). I replaced also the material at principal infinity-bundle with this polished discussion.

    • CommentRowNumber3.
    • CommentAuthorUrs
    • CommentTimeJan 8th 2011

    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 nnLab, but that is intentional: I want to collect in concise form everything relevant nicely on this page here, eventually).

    • CommentRowNumber4.
    • CommentAuthorUrs
    • CommentTimeJan 10th 2011
    • (edited Jan 10th 2011)

    I have further gone through the “Structures”-subsection:

    • I have parked some remarks on local over-(,1)(\infty,1)-toposes over small-projective objects XX and Π(X)\mathbf{\Pi}(X) at Paths and geometric Postnikov towers (I was going to use this to discsuss \infty-Lie algebroids over XX, 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.

    • CommentRowNumber5.
    • CommentAuthorUrs
    • CommentTimeJan 10th 2011
    • (edited Jan 10th 2011)

    I have made fully detailed the previously sketchy proof of the de Rham-adjunction (Π dR dR)(\mathbf{\Pi}_{dR} \dashv \mathbf{\flat}_{dR}) here.

    • CommentRowNumber6.
    • CommentAuthorUrs
    • CommentTimeJan 11th 2011
    • (edited Jan 11th 2011)

    I have added to cohesive oo-topos – de Rham cohomology a dicussion of the following:

    the objects Π dRX\mathbf{\Pi}_{dR} X in the cohesive \infty-topos H\mathbf{H} are connected previsely if in H\mathbf{H} the pieces have points - axiom holds in the sense that

    π 0ΓXπ 0ΠX \pi_0 \Gamma X \to \pi_0 \Pi X

    is an epi.

    This means that in the presence of this axiom, Π dR\mathbf{\Pi}_{dR} has essentially all the abstract properties of what is called the de Rham schematic homotopy type.

    That’s maybe noteworthy.

    • CommentRowNumber7.
    • CommentAuthorUrs
    • CommentTimeJan 11th 2011
    • (edited Jan 11th 2011)

    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 H\mathbf{H} be strongly \infty-connected:

    this implies that if one has a geometric homotopy fgf \Rightarrow g in H\mathbf{H} induced by a geometrically contractible interval object II – meaning that Π(I)\Pi(I) is sufficiently connected – then under Π:HGrpd\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(𝔤)\exp(\mathfrak{g}) at Lie integration).

    • CommentRowNumber8.
    • CommentAuthorUrs
    • CommentTimeJan 12th 2011

    have now gone through the section Structures in a cohesive oo-topos – Differential Cohomology and polished and expanded it.

    • CommentRowNumber9.
    • CommentAuthorUrs
    • CommentTimeJan 12th 2011
    • (edited Jan 12th 2011)

    re #6:

    I had defined “pieces have points” in a cohesive (,1)(\infty,1)-topos by demanding that π 0ΓXπ 0ΠX\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 ΓXΠX\Gamma X \to \Pi X being a regular epimorphism in an (infinity,1)-category (as now mentioned there). Which sounds much nicer.

    • CommentRowNumber10.
    • CommentAuthorUrs
    • CommentTimeJan 12th 2011

    I have now gone through the subsections

    • CommentRowNumber11.
    • CommentAuthorUrs
    • CommentTimeJan 12th 2011

    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.

    • CommentRowNumber12.
    • CommentAuthorUrs
    • CommentTimeJun 28th 2012
    • (edited Jun 28th 2012)

    I have added to cohesive (infinity,1)-topos – structures a new section Flat Ehresmann connections which discusses how for GG a cohesive \infty-group flat GG-Ehresmann \infty-connections can be understood as the twisted infinity-bundles for the local coefficient \infty-bundle

    dRBG BG BG. \array{ \flat_{dR} \mathbf{B}G &\to& \flat \mathbf{B}G \\ && \downarrow \\ && \mathbf{B}G } \,.