Holmstrom Quillen's Theorem B

arXiv:1101.4879 A Quillen theorem Bn for homotopy pullbacks from arXiv Front: math.CT by C. Barwick, D. M. Kan We prove an extension of the Quillen Theorem Bn for homotopy fibres to a similar result for homotopy pullbacks and use this to obtain sufficient conditions on a pullback diagram of categories to guarantee that it be a homotopy pullback.

Ref: Jardine-Goerss IV.5.2. It is stated there for a functor $F:C \to D$ between small cats satisfying some hyps, and concludes that a certain square involving BC and BD is homotopy cartesian. They also emphasize the following lemma: For a sSet-valued functor $X$ on a small cat $I$ sending all morphisms to weak equivalences, and for any object $j$ of $I$, the square given by $X(j)$, $d(holim(X))$, pt, $BI$ is homotopy cartesian. Also other variations on this theme.

nLab page on Quillen's Theorem B