TITLE: Good functorial model for BG
QUESTION [23 upvotes]: There are several functorial constructions of the space BG for a topological group (meaning BG plus the universal G-bundle). First, there is the Milnor construction, treated in several textbooks. The Milnor construction is functorial and $EG \to BG$ is locally trivial for all topological groups. The Milnor construction is NOT monoidal in the sense that $B(G \times H) \cong BH \times BG$ ($B1$ is something like an infinite-dimensional simplex and not a point).
On the other hand, there is the nerve-construction $BG:= |N_{\bullet} G|$ (plus a construction of $EG$). This is monoidal, but the map $EG \to BG$ is not always locally trivially (according to Graeme Segal, Classifying spaces and spectral sequences, p. 107). It is locally trivial if G is "locally well-behaved" (Segal gives a precise condition). Segal claims that if G is not locally well-behaved, then local triviality is not an appropriate concept.
I would be happy to exclude groups like the p-adic integers from having a classiying space, but there are other groups which I do not like to throw away, like Homeo (X) for a manifold X (is this locally well-behaved??). Here is my question:
Is there a construction of $BG$, satisfying the following properties:

functorial,
monoidal,
$EG \to BG$ is locally trivial,
the class of groups to which it applies is "very large", including Homeo of reasonable spaces,
Simple enough to be reasonably presented in a lecture course?

REPLY [5 votes]: Segal's classifying space $BG$ (the geometric realisation of the nerve of $G$ considered as a one-object topological groupoid) and the associated universal bundle (the geometric realisation of the nerve of the action groupoid of $G$ acting on itself by mulitplication, or equivalently, the codiscrete groupoid with objects $G$) was also studied by May, Milgram and Steenrod. These latter three only require that $G$ be well-pointed: the inclusion of the identity element is a closed cofibration. I note however that this is all done in the category of $k$-spaces.
The nice result of May is that $EG\to BG$ is not just a locally trivial bundle, but a numerable bundle, that is, there is a trivialising cover of $BG$ such that this cover admits a subordinate partition of unity. Thus $EG \to BG$ classifies numerable bundles (over paracompact spaces these are clearly the same as bundles).
Additionally, $EG$ is a topological group. Also since construction this uses ordinary geometric realisation, $E(-) \to B(-)$ preserves products (and indeed pullbacks) - this where the geometric realisation into k-spaces is used.