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1. • CommentRowNumber1.
   • CommentAuthorUrs
   • CommentTimeFeb 4th 2010
   • PermaLink
   Author: Urs
   Format: Markdownstarted [[monodromy]]

   started monodromy

2. • CommentRowNumber2.
   • CommentAuthorzskoda
   • CommentTimeFeb 4th 2010
   • PermaLink
   Author: zskoda
   Format: MarkdownNow I am confused about the higher homotopy groups aspect. Isn't it that in a sense higher analogues of universal covering spaces are played by higher Postnikov fibers ? Now monodromy is usually looked for pi-1 case, and it is clear to me why you go to infinity analogue right away. But I do not see how now levels for finite pi-k, or H-k (Hurewicz! once you are there) seen at finite levels are packed into the infinity picture ? Is it worthy to look that way or I have just baroque reminiscences of unwanted analogies ?

   Now I am confused about the higher homotopy groups aspect. Isn't it that in a sense higher analogues of universal covering spaces are played by higher Postnikov fibers ? Now monodromy is usually looked for pi-1 case, and it is clear to me why you go to infinity analogue right away. But I do not see how now levels for finite pi-k, or H-k (Hurewicz! once you are there) seen at finite levels are packed into the infinity picture ? Is it worthy to look that way or I have just baroque reminiscences of unwanted analogies ?

3. • CommentRowNumber3.
   • CommentAuthorUrs
   • CommentTimeFeb 4th 2010
   • PermaLink
   Author: Urs
   Format: MarkdownAhm, not sure. Maybe I don't quite understand what you have in mind. Did you look at Toen's article? He talks about this oo-monodromy group.

   Ahm, not sure. Maybe I don't quite understand what you have in mind.

   Did you look at Toen's article? He talks about this oo-monodromy group.

4. • CommentRowNumber4.
   • CommentAuthorzskoda
   • CommentTimeFeb 4th 2010
   • PermaLink
   Author: zskoda
   Format: MarkdownNever mind.

   Never mind.

5. • CommentRowNumber5.
   • CommentAuthordomenico_fiorenza
   • CommentTimeJun 28th 2012
   • PermaLink
   Author: domenico_fiorenza
   Format: MarkdownItexDoes a fibration $\pi:E\to X$ of (nice) topological spaces induce a fibration $\Pi(E)\to \Pi(X)$? (this shoud be obviously true or obviously false, but as worn out as I am at the moment I will leave it as a question). If that is true, then according to <a href="http://ncatlab.org/nlab/show/locally+constant+infinity-stack#ooStackOnTopSpace">Toen's equivalence</a>, one should have that $\pi:E\to X$ defines a [[local system]] on $X$. Moreover, since any map $f:Y\to X$ can be replaced by an equivalent fibration, one should have a good notion of (higher) monodromy defined for any morphism of topological spaces $f:Y\to X$.

   Does a fibration $\pi:E\to X$ of (nice) topological spaces induce a fibration $\Pi(E)\to \Pi(X)$? (this shoud be obviously true or obviously false, but as worn out as I am at the moment I will leave it as a question). If that is true, then according to Toen’s equivalence, one should have that $\pi:E\to X$ defines a local system on $X$. Moreover, since any map $f:Y\to X$ can be replaced by an equivalent fibration, one should have a good notion of (higher) monodromy defined for any morphism of topological spaces $f:Y\to X$.

6. • CommentRowNumber6.
   • CommentAuthorUrs
   • CommentTimeJun 28th 2012
   • (edited Jun 28th 2012)
   • PermaLink
   Author: Urs
   Format: MarkdownItex> Does a fibration π:E→X of (nice) topological spaces induce a fibration Π(E)→Π(X)? Yes, $\Pi = Sing$ is a right Quillen functor. See [here](http://ncatlab.org/nlab/show/homotopy%20hypothesis#ForKanComplexes). > one should have that π:E→X defines a local system on X. Yes. > Moreover, since any map f:Y→X can be replaced by an equivalent fibration, one should have a good notion of (higher) monodromy defined for any morphism of topological spaces f:Y→X. Yey, every morphism $E \to X$ with $\kappa$-small homotopy fibers is classified by a "monodromy map" $$ X \to \coprod_{F} B Aut(F) $$ where the coproduct ranges over $\kappa$-small homotopy types. If $X$ is connected we can restrict to one of these and have that $E \to X$ is classified by $$ X \to B Aut(F) \,. $$ This is originally a theorem of Stasheff and May. But it is also a simple instance of the general statement in section 4 of [NSSa](http://ncatlab.org/schreiber/show/Principal+%E2%88%9E-bundles+--+theory,+presentations+and+applications).

   Does a fibration π:E→X of (nice) topological spaces induce a fibration Π(E)→Π(X)?

   Yes, $\Pi = Sing$ is a right Quillen functor. See here.

   one should have that π:E→X defines a local system on X.

   Yes.

   Moreover, since any map f:Y→X can be replaced by an equivalent fibration, one should have a good notion of (higher) monodromy defined for any morphism of topological spaces f:Y→X.

   Yey, every morphism $E \to X$ with $\kappa$-small homotopy fibers is classified by a “monodromy map”

   $$X \to \coprod_{F} B Aut(F)$$

   where the coproduct ranges over $\kappa$-small homotopy types. If $X$ is connected we can restrict to one of these and have that $E \to X$ is classified by

   $$X \to B Aut(F) \,.$$

   This is originally a theorem of Stasheff and May. But it is also a simple instance of the general statement in section 4 of NSSa.

7. • CommentRowNumber7.
   • CommentAuthordomenico_fiorenza
   • CommentTimeJun 28th 2012
   • (edited Jun 28th 2012)
   • PermaLink
   Author: domenico_fiorenza
   Format: MarkdownItexThanks! I was interested in this since a linear representation of the monodromy map above is at the heart of the quantization map in section 8 of <a href="http://arxiv.org/abs/0905.0731/">Topological QFTs from compact Lie groups</a>

   Thanks! I was interested in this since a linear representation of the monodromy map above is at the heart of the quantization map in section 8 of Topological QFTs from compact Lie groups

8. • CommentRowNumber8.
   • CommentAuthorUrs
   • CommentTimeJul 2nd 2017
   • PermaLink
   Author: Urs
   Format: MarkdownItexI have added to _[[monodromy]]_ an elementary point-set discussion of the monodromy of covering spaces, [here](https://ncatlab.org/nlab/show/monodromy#InPointSetTopology).

   I have added to monodromy an elementary point-set discussion of the monodromy of covering spaces, here.