what does BG classify? i.e. what is a principal fibration?

Question

I'm looking for cold hard facts about just what $BG$ classifies, if $G$ is any grouplike topological monoid. I have some vague idea that $[X,BG]$ is in bijection with equivalence classes of "principal fibrations" over $X$. What exactly is a principal fibration?

May's Classifying Spaces and Fibrations looks like a good source, but I'm having trouble teasing out whether his notion of $G\mathcal U$-fibration (which he proves is classified by $BG$) is equivalent to the notion of a fibration $E \rightarrow B$ with a fiberwise right $G$-action giving weak equivalences $G \rightarrow E_b$, $g \mapsto xg$ for each point $x$ in the fiber $E_b$. (I can show the $\Rightarrow$ but not the $\Leftarrow$. Maybe there needs to be another condition in my naive description.)

Also, any grouplike monoid $G$ is weakly equivalent to $\Omega BG$, so "principal fibrations," whatever they are, should correspond (in some sense I want to make precise) to pullbacks of the path-loop fibration over $BG$.

Answer 1

In view of the references to my Memoir, Classifying spaces and fibrations, in other answers, I guess I should answer too. The requested answer is implicit but not quite explicit there. Fix a grouplike topological monoid $G$. Maybe assume for simplicity that its identity element is a nondegenerate basepoint (no loss of generality by 9.3). Define a $G$-torsor to be a right $G$-space $X$ such that the $G$-map $G\longrightarrow X$ that sends $g$ to $xg$ is a weak equivalence for $x\in X$. Let $\mathcal{G}$ be the category of $G$-torsors and maps of $G$-spaces between them. The Memoir defines a $\mathcal{G}$-fibration in terms of the $\mathcal{G}$-CHP, which is equivalent to having a $\mathcal{G}$-lifting function. Cary asks whether that notion is equivalent to an a priori weaker notion. The answer depends on what ``equivalence'' means. The Memoir insists on $\mathcal{G}$-fibrations, but it defines an equivalence (6.1) to be a $\mathcal{G}$-map over the base space, where a $\mathcal{G}$-map only has to be a map of $G$-torsors on fibers. (More precisely, it takes the equivalence relation generated by such maps.) Using $\mathcal{G}$-fibrations allows one often to replace that notion of equivalence by the nicer one of $\mathcal{G}$-fiber homotopy equivalence. But with the equivalence relation as given, any $\mathcal{G}$-map that is a quasifibration is equivalent to a $\mathcal{G}$-fibration, by the $\Gamma$-construction in Section 5. Therefore the classification theorem remains true allowing all $\mathcal{G}$-spaces that are quasifibrations, which of course includes Cary's preferred notion of a $G$-fibration.

Answer 2

When $G$ is discrete, another sort of answer is provided by the paper of Michael Weiss:

What does the classifying space of a category classify? Homology, Homotopy and Applications 7 (2005), 185–195.

Weiss shows that for a small category $C$, the classifying space $BC$ classifies sheaves of $C$–sets with representable stalks.

The equivalence relation on such sheaves over a space $X$ is given by concordance: given two such sheaves of the above kind, say $\cal F_0, \cal F_1$, one says they are concordant if there is a sheaf of the above kind over $X\times [0,1]$ which restricts to $\cal F_i$ on $X \times {i}$, $i \in {0,1}$.

I'm wondering if there's an analog of Weiss' result which holds for an arbitrary topological category. This would give a result for a general topological monoid.

Answer 3

This is not an answer but an attempt to clarify the question.

In the category of right $G$-spaces (with weak equivalences being the maps that as maps of spaces are weak equivalences) let us single out those objects $X$ for which there is a weak equivalence $G\to X$ (where $G$ has the usual right action). In other words, those such that for some $x$ the map $g\mapsto xg$ is a weak equivalence of spaces. If $G$ is grouplike then you can say "every" instead of "some" (as long as you remember to specify also that $X$ is not empty!).

Call these the "weak principal homogeneous spaces". Note that if $Y\to X$ is a weak equivalence of $G$-spaces and $X$ is of this kind, then $Y$ is as well.

I think the question might be something like this:

We would like to

(1) specify which weak principal homogeneous spaces will be allowed as fibers

(2) specify what we mean by "fibration" (locally trivial bundle? Serre fibration? quasifibration? ...)

and then consider as "principal $G$-fibrations" those maps $E\to B$ with fiber-preserving right $G$-action such the map is as in (2) and the fibers are as in (1), and then be able to say:

Homotopy classes of maps $B\to BG$ correspond bijectively with equivalence classes of principal $G$-fibrations on $B$. This requires that we also

(3) specify what we mean by an equivalence between two such principal fibrations on the same base.

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When $G$ is a topological group then the standard thing is to say (1) the fibers should be isomorphic to $G$ as $G$-spaces (2) locally trivial fiber bundle (and local triviality respecting the $G$-action follows), (3) isomorphism.

If you want to stick with "isomorphism" for (3) in the more general case, then:

Since there is only one homotopy class $\star\to BG$, for (1) you are committed to choosing a single $G$-space $X$ and allowing as fibers only things isomorphic to $X$.

And by considering bundles over disks it seems that you are also committed to choosing "locally trivial bundle" in (2) (there are locally trivializations respecting the $G$-action).

And it seems that this $X$ had better be such that its automorphism group, say $\Gamma$, is also weakly equivalent to $G$, or more precisely such that when considered as a left $\Gamma$-space $X$ is a weak principal homogeneous space.

It will not work to choose $G$ itself as $X$ unless the group of invertible elements of $G$ is equivalent to $G$.

You are in luck if, for example, $G$ is a topological monoid that happens to admit a weak equivalence $G\to \Gamma$ to a topological group such that $\Gamma$ is algebraically generated by the image. But this is rare.

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There is probably more then one right answer. Maybe you can allow all weak principal homogeneous spaces as fibers and use Serre fibrations (or maybe quasifibrations) and let equivalence between two such things over the base $B$ mean a map (respecting the map to $B$ and the $G$-action) that is a weak equivalence of total spaces, or equivalently of fibers. Does anybody know?

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Note: There is always a group $\Gamma$ related to $G$ indirectly by weak equivalences $G\leftarrow ? \to \Gamma$, so that $BG\simeq B\Gamma$ represents principle $\Gamma$-bundles, but I don't think that's the kind of answer that's wanted.