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Regarding "fibration" as a homotopy analogue of "fiber bundle",I want to see parallel notions of "structure group" and "fiber change" in "fibration".

Does it make sense to talk about "structure group" (or maybe "structure monoid") in fibration?

I guess such a notion should allow us to do "fiber change" for a fibration.

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    $\begingroup$ If $X$ is a finite CW complex, the space of self-homotopy equivalences $\text{Aut}(X)$ is a grouplike topological monoid. It's the appropriate notion of "structure group"; it admits a classifying space $B\text{Aut}(X)$, well-defined up to homotopy equivalence (even if you change the homotopy type of $X$). Fibrations with fiber $X$ are classified by maps into the above classifying space. This was all done, IIRC, by Stasheff, and then generalized by May to a larger class of $X$ - see May, "Classifying spaces and fibrations". $\endgroup$
    – mme
    Commented Apr 24, 2016 at 22:24
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    $\begingroup$ I'll add that you can always find, for every group-like topological monoid $M$, a model for $M$ that is an actual topological group (the usefulness of this fact is not very high, but it's cool to know). This is done, for example, in Quillen's On the group-completion of a simplicial monoid $\endgroup$
    – Denis Nardin
    Commented Apr 24, 2016 at 22:27
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    $\begingroup$ In particular, if $G$ is a closed subgroup of the homeomorphism group of $X$, then one has the map $BG\to BAut(X)$ and reduction of the structure group of the fibration to $G$ is a lift of the classifying map to $BG$. $\endgroup$
    – Igor Belegradek
    Commented Apr 25, 2016 at 3:36
  • $\begingroup$ Sorry, the above comment should say "even if you change $X$ to a homotopy equivalent complex"; of course you cannot change the homotopy type. Is it worth collecting these comments as one answer? $\endgroup$
    – mme
    Commented Apr 25, 2016 at 14:52
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    $\begingroup$ Mike, thanks for the reference. For others, his comment about larger X is that there is no need to restrict to finite CW complexes. That ensures that Aut(X) has the homotopy type of a CW complex, but the proofs in ``Classifying spaces and fibrations'' don't require that. Denis, if memory serves, I think that Milnor was the first to see that you can replace M by a topological group. $\endgroup$
    – Peter May
    Commented Apr 25, 2016 at 16:33

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My advisor (Sufian Husseini) wrote a book where something like this is done, based on the work in some of his first papers.

Husseini, S. Y. The topology of classical groups and related topics. Gordon and Breach Science Publishers, New York-London-Paris 1969 viii+128 pp.

His idea is to generalize the James construction to replace all of the usual fibration constructions with cellular maps, products and actions, using constructions of "reduced product type" (RPT). The main theorems have the general flavor of: "the standard theory of fiber bundles works for vibrations".

I think a more thorough modeling of fibrations by RPT methods was done by his Ph.D. student Hans Mathews in his thesis (University of Wisconsin, mid 1990s), but not published as far as I know.

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