A counterexample to a homotopy-equivalence criterion for topological categories

Roberts' Theorem 4.2 states that a fully faithful, essentially O_num-surjective functor between well-pointed topological categories induces a homotopy equivalence of classifying spaces. In the published formulation, however, the space of algebraically invertible arrows is only equipped with the subspace topology; continuity of inversion on that subspace is not assumed. We construct well-pointed topological categories X and Y and a fully faithful, essentially O_num-surjective functor f: X → Y for which the induced map Bf: BX → BY is not a homotopy equivalence. More precisely, H_1(BY; Q) is one-dimensional, whereas H_1(BX; Q) has dimension at least continuum. The counterexample isolates the missing continuity of inversion in the published argument. We also record an additional hypothesis—continuity of inversion on the subspace of invertible arrows—under which the continuity issue disappears and the remaining construction applies.