David Roberts
Theorem A for topological categories

A topological category XX is called well-pointed if the unit map X 0X 1X_0 \to X_1 is a fibrewise cofibration over X 0×X 0X_0 \times X_0. 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’.

Topological analogue of Quillen’s theorem A

On Top, let O nO_n be the pretopology of numerable open covers, and OO the pretopology of all open covers. An O nO_n-equivalence between topological categories where the codomain is well-pointed induces a homotopy equivalence between their classifying spaces. Note that if we assume Y 0Y_0 paracompact, any OO-equivalence in an O nO_n-equivalence, because numerable covers are cofinal in all open covers for paracompact spaces.

If we replace O nO_n by the pretopology of all open covers, then the geometric realization is a weak homotopy equivalence. If one uses fat realization, the assumption that YY 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 YY be a well-pointed topological category, XX a topological category, and f:XYf:X \to Y a continuous functor. If the map Bρ:B(Y 0f)Y 0B\rho:B(Y_0\downarrow f) \to Y_0 is shrinkable (resp. an acyclic Serre fibration), then Bf:BXBYBf:BX \to BY is a homotopy equivalence (resp. a weak homotopy equivalence).

Here the functor ρ:Y 0fY 0\rho:Y_0 \downarrow f \to Y_0 is the canonical functor from the comma category:

Y 0f X ρ f disc(Y 0) Y. \array{Y_0 \downarrow f & \rightarrow & X\\ \rho\downarrow & \Leftarrow& \downarrow f \\ disc(Y_0) & \hookrightarrow &Y.}

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.
