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 fiberwise closed cofibration over X 0×X 0X_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 Top\mathbf{Top}-enriched categories appeared in Vogt’s ‘Homotopy limits and colimits’.

Statement

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’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 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).

Last revised on November 14, 2020 at 23:03:02. See the history of this page for a list of all contributions to it.