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started monodromy
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 ?
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.
Never mind.
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 $f:Y\to X$.
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.
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
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