David Roberts
Theorem A for topological categories

A topological category $X$ is called well-pointed if the unit map $X_0\to X_1$ is a fibrewise cofibration over $X_0\times X_0$. 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 $O_n$ be the pretopology of numerable open covers, and $O$ the pretopology of all open covers. An $O_n$-equivalence between topological categories where the codomain is well-pointed induces a homotopy equivalence between their classifying spaces. Note that if we assume $Y_0$ paracompact, any $O$-equivalence in an $O_n$-equivalence, because numerable covers are cofinal in all open covers for paracompact spaces.

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

Here the functor $\rho :Y_0\downarrow f\to Y_0$ is the canonical functor from the comma category:

$$\begin{array}{ccc}{Y}_0\downarrow f& \to & X\\ \rho \downarrow & \Leftarrow & \downarrow f\\ \mathrm{disc}(Y_0)& \hookrightarrow & Y.\end{array}$$\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.