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We correct an error in [1], and provide a new shorter proof of Theorem B′. Introduction The purpose of this note is to point out that Lemma 1.5 of [1] is not true, but that a slightly weaker version is true and suffices for the applications in the paper. Lemma 1.5 is an ingredient in the proof of our analogues for simplicial sets of Theorems A and B of Quillen [3], which we called Theorems A′ and B′. Our proof followed the same general lines as Quillen’s, making use of his technique of constructing a quasifibration cell by cell. The problem with the proof in the paper is a cavalier application of the method of proof used in the lemma on page 90 of [3]; we ignored the degenerate cells in the skeletal filtration of the base space, or more precisely, the cells in the total space lying over them. Lemma 1.5′ below is a weakened form of Lemma 1.5 of [1] that provides a homotopy cartesian square rather than a quasifibration. This is enough for the proof of Theorem B′, which we reprove below using the new lemma. The proof of Lemma 1.5′ given here is completely combinatorial, and we thank the referee for dramatically shortening our original proof. A counterexample to the statement of Lemma 1.5 is provided by letting Y be the constant simplicial set with one point, and letting Z be some bisimplicial set such that all the face (and degeneracy) maps Z(A, ·) → Z(A′, ·) arising from maps A′ → A in ∆ are homotopy equivalences, and the map |Z([0], ·)| → |Z| is not a homeomorphism. Taking ρ ∈ Y ([0]) we see that |Zρ| = |Z([0], ·)| and F−1(ρ) = |Z|, and these two spaces are not homeomorphic, contrary to the claim in Lemma 1.5. Date: July 28, 2003. 2000 Mathematics Subject Classification. 19D06. Supported by NSF grants DMS 01-00587 and 99-70085. 1 2 GILLET AND GRAYSON Let ∆ be the category of finite nonempty ordered sets. For m ∈ ∆, let ∆ denote the simplicial set represented by m. For m ∈ N let [m] ∈ ∆ be the ordered set {0 < 1 < . . .m}, and let ∆ = ∆. Similarly, for a simplicial set X and m ∈ ∆ we’ll let Xm = X(m), and for m ∈ N we’ll let Xm = X([m]). Given simplicial sets X and Y , the external product X Y is the bisimplicial set defined by (X Y )m,n = Xm × Yn. Given a simplicial set Y define a bisimplicial set Y L by Y L(m,n) = Y (m), and define a bisimplicial set Y R by Y L(m,n) = Y (n). Observe that Y L ∼= Y ∆ and Y R ∼= ∆ Y . Lemma 1.5′. Suppose Z is a bisimplicial set, Y is a simplicial set, and F : Z → Y L is a map. For m ∈ ∆ and y ∈ Ym define a simplicial set Zy by Zy(n) = {z ∈ Zm,n | Fz = y}. Suppose that for all m ∈ ∆, all y ∈ Ym, and all g : p → m in ∆, the natural map Zy → Zg∗y defined by z 7→ (g 1)∗z is a homotopy equivalence. There is a map ∆ Zy → Z defined by (g, z) 7→ (g 1)∗z, and it fits into the following commutative square of bisimplicial sets. ∆ Zy // Z

Erratum to: ``The loop space of the $Q$-construction'' [Illinois J. Math. 31 (1987), no. 4, 574–597]