A topological category is called well-pointed if the unit map is a fibrewise cofibration over . In that case the nerve satisfies the Segal condition (originally described as being ‘good’ by Segal). If the topological category has one object, then the resulting topological monoid is well-pointed in the usual sense. Well-pointed Top-enriched categories appeared in Vogt’s ‘Homotopy limits and colimits’.
On Top, let be the pretopology of numerable open covers, and the pretopology of all open covers. An -equivalence between topological categories where the codomain is well-pointed induces a homotopy equivalence between their classifying spaces. Note that if we assume paracompact, any -equivalence in an -equivalence, because numerable covers are cofinal in all open covers for paracompact spaces.
If we replace by the pretopology of all open covers, then the geometric realization is a weak homotopy equivalence. If one uses fat realization, the assumption that is well-pointed can be dropped, but the best that can be achieved is a weak homotopy equivalence. This last version follows from a theorem in Moerdijk (appears in Springer’s LNM 1616, on page 63).
The well-pointed versions are corollaries from the following:
Quillen’s Theorem A for topological categories Let be a well-pointed topological category, a topological category, and a continuous functor. If the map is shrinkable (resp. an acyclic Serre fibration), then is a homotopy equivalence (resp. a weak homotopy equivalence).
Here the functor is the canonical functor from the comma category:
A topological version of Quillen B apparently first appeared in Meyer’s ‘Mappings of bar constructions’, but with what turns out in this application to be a different hypothesis.