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1. • CommentRowNumber1.
   • CommentAuthorUrs
   • CommentTimeJan 7th 2011
   • (edited Jan 7th 2011)
   • PermaLink
   Author: Urs
   Format: MarkdownItexI am now going through the section <a href="http://nlab.mathforge.org/nlab/show/cohesive+(infinity%2C1)-topos#Structures">Structures in a cohesive oo-topos</a> and polish and expand the discussions there. First thing I went through is the subsection <a href="http://nlab.mathforge.org/nlab/show/cohesive+(infinity%2C1)-topos#Homotopy">Geometric homotopy and Galois theory</a>. 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 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.)

2. • CommentRowNumber2.
   • CommentAuthorUrs
   • CommentTimeJan 7th 2011
   • (edited Jan 7th 2011)
   • PermaLink
   Author: Urs
   Format: MarkdownItexI have now gone through the section <a href="http://nlab.mathforge.org/nlab/show/cohesive+(infinity%2C1)-topos#Cohomology">Cohomology and principal oo-bundles</a>. 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.

   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.

3. • CommentRowNumber3.
   • CommentAuthorUrs
   • CommentTimeJan 8th 2011
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   Author: Urs
   Format: MarkdownItexat [[cohesive (infinity,1)-topos]] I have rewritten the words and expanded the content of the section <a href="http://ncatlab.org/nlab/show/cohesive+(infinity%2C1)-topos#ConcreteObjects">Concrete objects</a>. (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).

   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).

4. • CommentRowNumber4.
   • CommentAuthorUrs
   • CommentTimeJan 10th 2011
   • (edited Jan 10th 2011)
   • PermaLink
   Author: Urs
   Format: MarkdownItexI 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 <a href="http://nlab.mathforge.org/nlab/show/cohesive+(infinity%2C1)-topos#Paths">Paths and geometric Postnikov towers</a> (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 <a href="http://nlab.mathforge.org/nlab/show/cohesive+(infinity%2C1)-topos#FlatDifferentialCohomology">Flat $\infty$-connections and local systems</a>. 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 <a href="http://nlab.mathforge.org/nlab/show/cohesive+(infinity%2C1)-topos#deRhamCohomology">de Rham cohomology</a> 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 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.

5. • CommentRowNumber5.
   • CommentAuthorUrs
   • CommentTimeJan 10th 2011
   • (edited Jan 10th 2011)
   • PermaLink
   Author: Urs
   Format: MarkdownItexI have made fully detailed the previously sketchy proof of the de Rham-adjunction $(\mathbf{\Pi}_{dR} \dashv \mathbf{\flat}_{dR})$ <a href="http://nlab.mathforge.org/nlab/show/cohesive%20(infinity,1)-topos#DeRhamAdjunction">here</a>.

   I have made fully detailed the previously sketchy proof of the de Rham-adjunction here.

6. • CommentRowNumber6.
   • CommentAuthorUrs
   • CommentTimeJan 11th 2011
   • (edited Jan 11th 2011)
   • PermaLink
   Author: Urs
   Format: MarkdownItexI have added to <a href="http://ncatlab.org/nlab/show/cohesive+(infinity%2C1)-topos#deRhamCohomology">cohesive oo-topos -- de Rham cohomology</a> 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 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.

7. • CommentRowNumber7.
   • CommentAuthorUrs
   • CommentTimeJan 11th 2011
   • (edited Jan 11th 2011)
   • PermaLink
   Author: Urs
   Format: MarkdownItexI have added to the section <a href="http://nlab.mathforge.org/nlab/show/cohesive+(infinity%2C1)-topos#Homotopy">Geometric homotopy and Galois theory</a> 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]]).

   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).

8. • CommentRowNumber8.
   • CommentAuthorUrs
   • CommentTimeJan 12th 2011
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   Author: Urs
   Format: MarkdownItexhave now gone through the section <a href="http://nlab.mathforge.org/nlab/show/cohesive+(infinity%2C1)-topos#DifferentialCohomology">Structures in a cohesive oo-topos -- Differential Cohomology</a> and polished and expanded it.

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

9. • CommentRowNumber9.
   • CommentAuthorUrs
   • CommentTimeJan 12th 2011
   • (edited Jan 12th 2011)
   • PermaLink
   Author: Urs
   Format: MarkdownItexre #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.

   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.

10. • CommentRowNumber10.
    • CommentAuthorUrs
    • CommentTimeJan 12th 2011
    • PermaLink
    Author: Urs
    Format: MarkdownItexI have now gone through the subsections * <a href="http://nlab.mathforge.org/nlab/show/cohesive+(infinity%2C1)-topos#ChernWeilTheory">Chern-Weil homomorphisms and oo-connections</a> and have polished and slightly expanded it; * <a href="http://nlab.mathforge.org/nlab/show/cohesive+(infinity%2C1)-topos#ChernSimonsTheory">Higher holonomy and oo-ChernSimons theory</a> 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.

    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.

11. • CommentRowNumber11.
    • CommentAuthorUrs
    • CommentTimeJan 12th 2011
    • PermaLink
    Author: Urs
    Format: MarkdownItexokay, 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 * <a href="http://nlab.mathforge.org/nlab/show/cohesive+(infinity%2C1)-topos#ChernSimonsTheory">Higher holonomy and Chern-Simons *functional*</a> which is more accurate.

    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

    • Higher holonomy and Chern-Simons functional

    which is more accurate.

12. • CommentRowNumber12.
    • CommentAuthorUrs
    • CommentTimeJun 28th 2012
    • (edited Jun 28th 2012)
    • PermaLink
    Author: Urs
    Format: MarkdownItexI have added to [[cohesive (infinity,1)-topos -- structures]] a new section [Flat Ehresmann connections](http://ncatlab.org/nlab/show/cohesive+%28infinity%2C1%29-topos+--+structures#FlatEhresmannConnections) 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 $$ \array{ \flat_{dR} \mathbf{B}G &\to& \flat \mathbf{B}G \\ && \downarrow \\ && \mathbf{B}G } \,. $$

    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