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nLab > Latest Changes: structures in a cohesive oo-topos

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- 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 $\infty$ -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.)
- 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 $G$ -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.
- 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 $n$ Lab, 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)
- 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- $(\infty,1)$ -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.
- 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 $(\mathbf{\Pi}_{dR} \dashv \mathbf{\flat}_{dR})$ here.
- 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 $\mathbf{\Pi}_{dR} X$ 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.
- 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 $\infty$ -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).
- CommentRowNumber8.
- CommentAuthorUrs
- CommentTimeJan 12th 2011
- PermaLink
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.
- 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 $(\infty,1)$ -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.
- 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 $\infty Lie Grpd$ 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.
- 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.
- 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 $G$ a cohesive $\infty$ -group flat $G$ -Ehresmann $\infty$ -connections can be understood as the twisted infinity-bundles for the local coefficient $\infty$ -bundle
$\array{ \flat_{dR} \mathbf{B}G &\to& \flat \mathbf{B}G \\ && \downarrow \\ && \mathbf{B}G } \,.$

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nLab > Latest Changes: structures in a cohesive oo-topos