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 fiberwise closed cofibration over $X_0 \times X_0$. In that case the nerve satisfies the condition that all of its degeneracy maps are closed cofibrations (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 $\mathbf{Top}$-enriched categories appeared in Vogt’s ‘Homotopy limits and colimits’.

Statement

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 $Bf:BX \to BY$ 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:

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

A topological version of Quillen’s Theorem B apparently first appeared in Meyer’s ‘Mappings of bar constructions’, but with what turns out in this application to be a different hypothesis.

Corollaries

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).

Published version

The final version of the theorem appears in:

• Homotopy types of topological stacks of categories, New York Journal of Mathematics, Volume 30 (2024), 940-955, journal version, arXiv:2204.02778