KAN SUBDIVISION AND PRODUCTS OF SIMPLICIAL SETS

VEGARD FJELLBO AND JOHN ROGNES

Abstract. The canonical map from the Kan subdivision of a product of finite simplicial sets to the product of the Kan subdivisions is a simple map, in the sense that its geometric realization has contractible point inverses.

1. Introduction

Kan’s normal subdivision [5, §7] is a functor Sd from simplicial sets to simplicial sets. It agrees with barycentric subdivision when applied to (ordered) simplicial complexes. The maps induced by applying Kan subdivision to the projections X × Y → X and X × Y → Y combine to a canonical map κ: Sd(X × Y) → Sd X × Sd Y.

Definition 1.1. A map f: A → B of finite simplicial sets is said to be simple [9, 2.1.1] if its geometric realization |f|: |A| → |B| has contractible point inverses, meaning that the preimage |f|^{−1}(b) is contractible for each point b ∈ |B|. We write A ⟶^{≃_{s}} B to denote a simple map.

Theorem 1.2. Let X and Y be finite simplicial sets. The canonical map

κ: Sd(X × Y) ⟶^{≃_{s}} Sd X × Sd Y ,

from the Kan subdivision of the product X×Y to the product of the Kan subdivisions, is a simple map.

The theorem follows from the special cases when X = Δ[m] is a standard simplex, for some m ≥ 0, by an induction on the dimension and number of topdimensional cells in the CW complex |X|. A second induction, over the cells of |Y |, allows us to specialize further to the cases when Y = Δ[n], for some n ≥ 0. Hence our real task is to prove the following result.

Proposition 1.3. The canonical map

κ: Sd(Δ[m] × Δ[n]) ⟶^{≃_{s}} Sd Δ[m] × Sd Δ[n]

is simple, for each m ≥ 0 and n ≥ 0.

Example 1.4. For m = n = 1, we have the following picture.

The map κ is the identity on the boundary, and takes the three interior vertices on the left hand side to the single interior vertex on the right hand side. The rhombus on the left is collapsed to the diagonal on the right. The point inverses of |κ| are points or closed intervals. Hence |κ| is not a homeomorphism, but all preimages of points are contractible. Note that |κ| does not admit a continuous section.

Date: June 6th 2014.

2010 Mathematics Subject Classification. 55U10, 57Q10.

Remark 1.5. Simple maps have a direction: given a simple map f: A → B there might be no simple map g: B → A. Simple-homotopy theory [2] concerns the equivalence relation on finite simplicial sets (or simplicial complexes) generated by simple maps. Piecewise-linear topology [6] concerns the properties of simplicial complexes that are invariant under linear subdivisions. It is often convenient and sufficient to only consider the linear subdivisions that arise by iterated barycentric subdivision [7, §3.5, §3.6]. Our theorem gives a strong version of the statement that the simple-homotopy type of a product is a well-defined piecewise-linear notion, also with respect to this more restricted notion of subdivision.

The weaker statement that the map κ in the theorem is a simple-homotopy equivalence is easily proved: there is a natural last vertex map d_{X}: Sd X → X, which is simple for all finite X [9, 2.2.17], and (d_{X} × d_{Y}) ∘ κ = d_{X×Y}. However, this argument does not suffice to prove that κ is a simple map.

In Section 2 we present an application of our main result. In Section 3 we show how to deduce Theorem 1.2 from Proposition 1.3, and outline the proof of the latter result. The details of that proof occupy Sections 4, 5 and 6.

2. An application to the improvement functor

A simplicial set X is said to be non-singular [9, 1.2.2] if for each non-degenerate nsimplex x in X the representing map x̄: Δ[n] → X is a cofibration, or equivalently, if each non-degenerate n-simplex has n + 1 distinct vertices, for each n ≥ 0. This implies that the geometric realization |X| has a preferred piecewise-linear structure. Let 𝒞 be the category of finite simplicial sets and simplicial maps, let 𝒟 be its full subcategory of finite non-singular simplicial sets, and let s𝒞 and s𝒟 denote the respective subcategories of simple maps.

Our main result has an application to the multiplicative properties of the improvement functor I: 𝒞 → 𝒟 constructed in [9, 2.5.2]. This functor associates to each finite simplicial set X a finite non-singular simplicial set I(X), together with a natural simple map s_{X}: I(X) → X. Hence the restricted functor sI: s𝒞 → s𝒟 induces a homotopy equivalence of classifying spaces, and the localized functor I[s^{−1}]: 𝒞[s^{−1}] → 𝒟[s^{−1}] (inverting the respective collections of simple maps [4, 1.1]) is an equivalence of categories. This shows that the simple-homotopy theory of finite simplicial sets is equivalent to the simple-homotopy theory of finite non-singular simplicial sets. For a useful relative version of this statement, see [9, 1.2.5]. The application we have in mind concerns the compatibility of this equivalence with the categorical products.

Proposition 2.1. Let X and Y be finite simplicial sets, and let I denote the improvement functor from [9, §2.5]. The canonical map

I(X × Y) ⟶^{≃_{s}} I(X) × I(Y) ,

induced by the projections X × Y → X and X × Y → Y, is simple.

Before showing how to deduce this from our main theorem, we recall some notation and terminology. Each simplex x of a simplicial set X is a degeneration of a unique non-degenerate simplex, which we denote x^{#}. If x is non-degenerate then x = x^{#}. Let X^{#} be the set of non-degenerate simplices in X, partially ordered by letting x ≤ y if x is a face of y. We call the nerve B(X) = N(X^{#}) of this partially ordered set the Barratt nerve of X, due to its early appearance in [1]. A simplicial map f: X → Y induces an order-preserving function f^{#} : X^{#} → Y^{#}, taking x to f(x)^{#}, and a functorial map of nerves B(f) = N(f^{#}): B(X) → B(Y). The Kan subdivision Sd(X) is defined as the left Kan extension of [n] ↦ B(Δ^{n}) along the Yoneda embedding of Δ into simplicial sets [9, 2.2.7], so there is a natural map b_{X}: Sd(X) → B(X), which is an isomorphism for non-singular X [9, 2.2.11]. The improvement functor I is defined as the composite

I(X) = B(Sd(X)) .

(The use of the opposite subdivision in [9] plays no explicit role in our arguments, and will be suppressed.)

Remark 2.2. The improvement functor I appears implicitly in the work of Thomason [8], as the composite N ∘ c Sd^{2} of the nerve functor N (from small categories to simplicial sets) and the categorified double subdivision c Sd^{2} (from simplicial sets to categories). The latter functor is the left adjoint in Thomason’s Quillen equivalence between these two (closed) model categories. By [8, Lemma 5.6] the category c Sd^{2}(X) is a partially ordered set, for any simplicial set X, and this makes it easy to identify it with the partially ordered set Sd(X)^{#} of non-degenerate simplices in Sd(X).

Proof of Proposition 2.1. There are canonical maps

Sd(Sd(X × Y)) → Sd(Sd(X) × Sd(Y)) → Sd(Sd(X)) × Sd(Sd(Y)) .

The left hand map, Sd(κ), is simple by our Theorem 1.2, combined with the fact that Kan subdivision preserves simple maps [9, 2.3.3]. The right hand map is simple by the same theorem applied for the finite simplicial sets Sd(X) and Sd(Y). Hence the composite map is also simple [9, 2.1.3(a)].

The natural map b_{Sd(X)} : Sd(Sd(X)) → B(Sd(X)) is simple by [9, 2.5.5, 2.5.8]. Hence the vertical maps are simple in the commutative square

\xymatrix{Sd(Sd(X × Y)) \ar[r]^{≃_{s}} \ar[d]_{≃_{s}} & Sd(Sd(X)) × Sd(Sd(Y)) \ar[d]^{≃_{s}} \\ B(Sd(X × Y)) \ar[r] & B(Sd(X)) × B(Sd(Y)) .}

We have just argued that the upper horizontal map is simple, and this implies that the lower horizontal map is simple, by the right cancellation property of simple maps [9, 2.1.3(b)]. □

3. Outline of argument

Proof of Theorem 1.2 assuming Proposition 1.3. To reduce Theorem 1.2 to the case X = Δ[m], we write X = Δ[m]∪_{∂Δ[m]}X′ and think of κ: Sd(X×Y) → Sd X×Sd Y as the vertical map of horizontal pushouts in the diagram

[figure: Sd(Δ[m] × Y); κ; Sd(∂Δ[m] × Y); κ; Sd(X′ × Y); κ; Sd Δ[m] × Sd Y; Sd ∂Δ[m] × Sd Y; Sd X′ × Sd Y .]

This uses the fact that Sd preserves colimits and cofibrations [9, 2.2.9]. By induction we may assume that κ is simple for ∂Δ[m] and X′. The case of general X then follows from the case X = Δ[m], by the gluing lemma for simple maps [9, 2.1.3(d)].

The reduction to the case Y = Δ[n] is similar. Write Y = Δ[n] ∪_{∂Δ[n]} Y′ and think of κ: Sd(Δ[m]×Y) → Sd Δ[m]×Sd Y as the map of pushouts in the diagram

[figure: Sd(Δ[m] × Δ[n]); κ; Sd(Δ[m] × ∂Δ[n]); κ; Sd(Δ[m] × Y′); κ; Sd Δ[m] × Sd Δ[n]; Sd Δ[m] × Sd ∂Δ[n]; Sd Δ[m] × Sd Y′ .]

By induction we may assume that κ is simple for ∂Δ[n] and Y′, so the main theorem follows from the case Y = Δ[n], i.e., from Proposition 1.3. □

Outline of proof of Proposition 1.3. To show that the canonical map

κ: Sd(Δ[m] × Δ[n]) ⟶ Sd Δ[m] × Sd Δ[n]

is simple, it suffices to show that it is simple over the interior of each non-degenerate r-simplex (z, w) = (z_{0} ≤ ⋯ ≤ z_{r}, w_{0} ≤ ⋯ ≤ w_{r}) of the target, for r ≥ 0, cf. Notation 4.2. By Lemmas 4.3, 4.7 and 4.11, κ agrees with the nerve of the order-preserving function π: P (ϕ_{r}, …, ϕ_{1}) → [r] over that interior. Here

F_{r} ⟶^{ϕ_{r}} … ⟶^{ϕ_{1}} F_{0}

is a diagram of partially ordered sets and order-preserving functions, specified in Notations 4.6 and 4.8, and P (ϕ_{r}, …, ϕ_{1}) = F_{r} ⊔_{ϕ_{r}} ⋯⊔_{ϕ_{1}} F_{0} is as in Definition 4.10.

Let

NF_{r} ⟶^{f_{r}} … ⟶^{f_{1}} NF_{0}

be the induced diagram of nerves. According to Definition 5.4, the nerve of π equals the reduced coordinate projection π: M(f_{r}, …, f_{1}) → Δ[r], where M(f_{r}, …, f_{1}) is the r-fold iterated reduced mapping cylinder. As we explain in Remark 5.11, there is a commutative diagram

\xymatrix{T (f_{r}, …, f_{1}) \ar[r]^{red} \ar[d] & M(f_{r}, …, f_{1}) \ar[d]^{π} \\ T^{r} \ar[r]^{red} & Δ[r] ,}

where T (f_{r}, …, f_{1}) is the r-fold iterated ordinary mapping cylinder, cf. Definition 5.6. The reduction map red: T (f_{r}, …, f_{1}) → M(f_{r}, …, f_{1}) is simple by Lemma 5.18, and the map red: T^{r} → Δ[r] is simple by Lemma 5.10.

By the composition and right cancellation properties of simple maps [9, 2.1.3], the reduced coordinate projection π will be simple if the ordinary cylinder projection T (f_{r}, …, f_{1}) → T^{r} is simple. Each point inverse of the geometric realization |T (f_{r}, …, f_{1})| → |T^{r}| is homeomorphic to one of the spaces |NF_{i}|, for 0 ≤ i ≤ r, so it suffices to prove that each of the partially ordered sets F_{i} has contractible classifying space. Using Notation 6.1, each F_{i} is of the form P^{z′,w′} for a suitable i-simplex (z′, w′) of Sd Δ[m] × Sd Δ[n]. Hence the required contractibility is a consequence of our final technical result, Proposition 6.7. □

4. Partially ordered sets of paths

As a first step towards the proof of Proposition 1.3, we unravel the definition of the canonical map κ in that special case, and identify the part of κ that sits over a non-degenerate r-simplex in its target.

The m-simplex Δ[m] = N([m]) is the nerve of the totally ordered set [m] = {0 < 1 < ⋯ < m}. Its k-simplices are the order-preserving functions [k] → [m], which we will refer to as operators, following [3, §4.1]. Injective operators are called face operators, and surjective operators are called degeneracy operators. The nondegenerate simplices of Δ[m] are the face operators μ: [k] → [m], which correspond to the non-empty subsets im(μ) of [m]. As in Section 2, we partially order the set Δ[m]^{#} of non-degenerate simplices by setting μ ≤ ζ if μ is a face of ζ, or equivalently, if the image of μ is a subset of the image of ζ. The Kan subdivision Sd Δ[m] = N(Δ[m]^{#}) is the nerve of that partially ordered set. Hence an rsimplex of Sd Δ[m] is a chain ζ_{0} ≤ ζ_{1} ≤ ⋯ ≤ ζ_{r} of face operators ζ_{i}: [k_{i}] → [m] for 0 ≤ i ≤ r.

The product Δ[m] × Δ[n] can be regarded as the nerve of [m] × [n] with the product partial ordering. Its k-simplices are the order-preserving functions [k] → [m]×[n], and its non-degenerate k-simplices are the injective order-preserving functions

γ: [k] → [m] × [n] ,

which correspond to the non-empty totally ordered subsets im(γ) of [m] × [n]. The set C = (Δ[m] × Δ[n])^{#} of non-degenerate simplices is partially ordered by setting β ≤ γ if β is a face of γ, or equivalently, if the image of β is a subset of the image of γ. The product Δ[m] × Δ[n] is a non-singular simplicial set, so its Kan subdivision Sd(Δ[m] × Δ[n]) = N((Δ[m] × Δ[n])^{#}) is the nerve of the partially ordered set of its non-degenerate simplices. An r-simplex is a chain γ_{0} ≤ γ_{1} ≤ ⋯ ≤ γ_{r} of injective order-preserving functions γ_{i}: [k_{i}] → [m] × [n] for 0 ≤ i ≤ r.

The first projection pr_{1} : Δ[m] × Δ[n] → Δ[m] induces the order-preserving function pr_{1}^{#} that takes each injective order-preserving function γ: [k] → [m] × [n] to the non-degenerate part (pr_{1} ∘γ)^{#} : [m_{1}] → [m] of the composite pr_{1} ∘γ. This is the non-degenerate simplex with image im(pr_{1}^{#}(γ)) = im(pr_{1} ∘γ). Similarly the second projection pr_{2} : Δ[m] × Δ[n] → Δ[n] induces an order-preserving function pr_{2}^{#} that takes γ to the non-degenerate simplex [n_{1}] → [n] with image im(pr_{2} ∘γ). We can display these order-preserving functions in the following diagram, where a feathered arrow denotes an injection and a two-headed arrow denotes a surjection.

[figure: [m_{1}]; pr_{1}^{#}(γ); [k]; γ; [n_{1}]; pr_{2}^{#}(γ); [m]; [m] × [n]; _{pr_{1}}; _{pr_{2}}; [n]]

The functorial map Sd(pr_{1}): Sd(Δ[m]×Δ[n]) → Sd Δ[m] equals the map of nerves N(pr_{1}^{#}) induced by pr_{1}^{#}, and similarly for the second projection. Hence the canonical map κ = (Sd(pr_{1}), Sd(pr_{2})) equals the map (N(pr_{1}^{#}), N(pr_{2}^{#})), which in turn can be identified with the map of nerves induced by

(pr_{1}^{#}, pr_{2}^{#}): (Δ[m] × Δ[n])^{#} → Δ[m]^{#} × Δ[n]^{#} ,

since the diagram

\xymatrix{N((Δ[m] × Δ[n])^{#}) \ar[d]_{N(pr_{1}^{#},pr_{2}^{#})} \ar[dr]^{κ} \\ N(Δ[m]^{#} × Δ[n]^{#}) \ar[r]_{≅} & N(Δ[m]^{#}) × N(Δ[n]^{#})}

commutes.

Definition 4.1. Let μ ∈ Δ[m]^{#} and ν ∈ Δ[n]^{#} be non-degenerate simplices. A non-degenerate simplex γ ∈ (Δ[m] × Δ[n])^{#} maps under (pr_{1}^{#}, pr_{2}^{#}) to (μ, ν) if and only if im(pr_{1} ∘γ) = im(μ) and im(pr_{2} ∘γ) = im(ν). In this case we say that γ is a (μ, ν)-path. We write P^{μ,ν} for the set of (μ, ν)-paths, partially ordered as a subset of C = (Δ[m] × Δ[n])^{#}.

This terminology (being a (μ, ν)-path) will be important throughout this paper. Note that being a (μ, ν)-path is a stronger condition than just being a path with image in im(μ) × im(ν).

Notation 4.2. Let (z, w) = (z_{0} ≤ ⋯ ≤ z_{r}, w_{0} ≤ ⋯ ≤ w_{r}) be a non-degenerate r-simplex in Sd Δ[m] × Sd Δ[n], with characteristic map \overline{(z, w)}: Δ[r] → Sd Δ[m] × Sd Δ[n], so that z_{i} ∈ Δ[m]^{#} and w_{i} ∈ Δ[n]^{#} are non-degenerate simplices for 0 ≤ i ≤ r. Let P_{i} = P^{z_{i},w_{i}} denote the partially ordered set of (z_{i}, w_{i})-paths.

The (z_{i}, w_{i}) are pairwise distinct, so the P_{i} are pairwise disjoint as sets, for 0 ≤ i ≤ r. Let P_{0} ∪ ⋯ ∪ P_{r} denote the union of these sets, partially ordered as a subset of C = (Δ[m]×Δ[n])^{#}, and let λ: P_{0} ∪⋯∪P_{r} → [r] be the order-preserving function taking the elements of P_{i} to i, for each 0 ≤ i ≤ r.

By the restriction of a map f: A → B over a simplicial subset Z ⊆ B, we mean the associated map f^{−1}(Z) → Z, given by restricting f to f^{−1}(Z) ⊆ A, and corestricting the target to Z.

Lemma 4.3. The restriction of κ over the simplicial subset of Sd Δ[m] × Sd Δ[n] generated by (z, w) is isomorphic to the map Nλ: N(P_{0} ∪ ⋯ ∪ P_{r}) → Δ[r].

Proof. Nerves commute with pullbacks, which means we have a pullback square

[figure: N(P_{0} ∪ ⋯ ∪ P_{r}); Nλ; Sd(Δ[m] × Δ[n]); κ; Δ[r]; \overline{(z,w)}; Sd Δ[m] × Sd Δ[n] .]

□

Corollary 4.4. Let (μ, ν) be a 0-simplex in Sd Δ[m] × Sd Δ[n]. The preimage under κ of the simplicial subset generated by (μ, ν) equals the nerve of the partially ordered set P^{μ,ν} of (μ, ν)-paths. Hence the preimage under |κ| of the corresponding 0-cell in | Sd Δ[m] × Sd Δ[n]| equals the classifying space |NP^{μ,ν}| of that partially ordered set.

We now explain how only a part of the union P_{0} ∪ ⋯ ∪ P_{r} plays a role over the interior of the r-cell |Δ[r]|.

Definition 4.5. Let (μ, ν), (ζ, η) ∈ Δ[m]^{#} × Δ[n]^{#} be such that μ ≤ ζ and ν ≤ η, and let L = im(μ) × im(ν) ⊆ [m] × [n]. We say that a (ζ, η)-path γ: [k] → [m] × [n] is (μ, ν)-full if some face β of γ is a (μ, ν)-path. In this case there is a greatest such face β, with image im(β) = im(γ) ∩ L. We write β = γ ∩ L for this greatest (μ, ν)-path.

This terminology (being (μ, ν)-full) will also be important throughout this paper. In general, not every (ζ, η)-path will be (μ, ν)-full.

Notation 4.6. For each 0 ≤ i ≤ r, let F_{i} denote the set of (z_{i}, w_{i})-paths that are (z_{j}, w_{j})-full for every 0 ≤ j < i. Partially order F_{i} as a subset of P_{i}, and partially order F_{0} ∪ ⋯ ∪ F_{r} as a subset of P_{0} ∪ ⋯ ∪ P_{r}. The F_{i} are pairwise disjoint as sets, since this holds for the P_{i}. Let ξ: F_{0} ∪ ⋯ ∪ F_{r} → [r] be the order-preserving function that takes each element of F_{i} to i.

The notations P_{i} and F_{i} are chosen for their brevity. They always depend on the implicit choice of a non-degenerate r-simplex (z, w), as above.

Lemma 4.7. The maps |Nξ| and |Nλ| agree over the interior of |Δ[r]|.

Proof. By the definition of ξ and λ, the triangle

[figure: N(F_{0} ∪ ⋯ ∪ F_{r}); Nξ; N(P_{0} ∪ ⋯ ∪ P_{r}); Nλ; Δ[r]]

commutes. Let x ∈ N(P_{0} ∪ ⋯ ∪ P_{r}) be a p-simplex that does not lie over the boundary ∂Δ[r] of Δ[r]. In other words, x is an order-preserving function such that the composite

[p] ⟶^{x} P_{0} ∪ ⋯ ∪ P_{r} ⟶^{λ} [r]

is surjective. We claim that x factors through F_{0} ∪ ⋯ ∪ F_{r}. To prove this, let u ∈ [p] be arbitrary, fix i so that x(u) ∈ P_{i}, and consider any 0 ≤ j < i. The surjectivity of λ ∘ x tells us that there is a v ∈ [p] with x(v) ∈ P_{j}. We cannot have v ≥ u, since λ ∘ x is order-preserving, so v < u and the (z_{j}, w_{j})-path x(v) must be a face of x(u). This shows that x(u) is (z_{j}, w_{j})-full, which implies that x(u) ∈ F_{i}.

It follows that |N(F_{0} ∪ ⋯ ∪ F_{r})| → |N(P_{0} ∪ ⋯ ∪ P_{r})| restricts to the identity over the interior of |Δ[r]|. □

Notation 4.8. Let L_{i} = im(z_{i})×im(w_{i}), for each 0 ≤ i ≤ r, and let ϕ_{i}: F_{i} → F_{i−1} be the order-preserving function γ ↦ γ ∩ L_{i−1}, for each 0 < i ≤ r.

As before, these notations depend on the implicit choice of a non-degenerate r-simplex (z, w). To conclude this section, we clarify how the partially ordered set F_{0} ∪ ⋯ ∪ F_{r} is determined by the diagram

F_{r} ⟶^{ϕ_{r}} … ⟶^{ϕ_{1}} F_{0}

of partially ordered sets and order-preserving functions.

Definition 4.9 ([9, 2.4.3]). Given an order-preserving function φ: V → W of partially ordered sets, let P (φ) = V ⊔_{φ} W be the disjoint union of the sets V and W, with the partial ordering generated by the relations in V, the relations in W, and the relation φ(v) < v for all v ∈ V.

In other words, for v ∈ V and w ∈ W we have w ≤ v in P (φ) if and only if w ≤ φ(v) in W. We write φ ∨ 1: P (φ) → W for the order-preserving function that restricts to φ on V and to the identity on W, and π: P (φ) → [1] = {0 < 1} for the order-preserving function that maps V to 1 and W to 0.

Definition 4.10 ([9, 2.4.13]). For a sequence of r order-preserving functions

V_{r} ⟶^{φ_{r}} … ⟶^{φ_{1}} V_{0}

let P (φ_{r}, …, φ_{1}) = V_{r} ⊔_{φ_{r}} ⋯ ⊔_{φ_{1}} V_{0} be the disjoint union of the sets V_{0}, …, V_{r}, with the partial ordering generated by the relations in V_{i} for 0 ≤ i ≤ r, and the relations φ_{i}(v) < v for all v ∈ V_{i} and 0 < i ≤ r. We write π: P (φ_{r}, …, φ_{1}) → [r] for the order-preserving function that takes V_{i} to i, for each 0 ≤ i ≤ r.

Alternatively, for r ≥ 2 we can construct P (φ_{r}, …, φ_{1}) as an instance P (ψ_{1}) of the previous definition, by an induction on the length of the sequence of functions. To start the induction, let ψ_{r} = φ_{r}. Thereafter, for 1 ≤ j < r, let ψ_{j} be the composite

P (φ_{r}, …, φ_{j+1}) ⟶^{ψ_{j+1}∨1} V_{j} ⟶^{φ_{j}} V_{j−1} .

Then P (φ_{r}, …, φ_{j}) = P (ψ_{j}) as partially ordered sets.

Lemma 4.11. Let F_{r} ⟶^{ϕ_{r}} … ⟶^{ϕ_{1}} F_{0} be as in Notation 4.8. Then F_{0} ∪ ⋯ ∪ F_{r} = P (ϕ_{r}, …, ϕ_{1}) as partially ordered sets, and the functions ξ: F_{0} ∪ ⋯ ∪ F_{r} → [r] and π: P (ϕ_{r}, …, ϕ_{1}) → [r] coincide.

Proof. The disjoint unions F_{0} ∪ ⋯ ∪ F_{r} and F_{r} ⊔_{ϕ_{r}} ⋯ ⊔_{ϕ_{1}} F_{0} agree as subsets of C = (Δ[m] × Δ[n])^{#}. The lemma asserts that the subset partial ordering on F_{0} ∪ ⋯ ∪ F_{r} inherited from C equals the partial ordering generated by the subset partial orderings on the individual subsets F_{i} for 0 ≤ i ≤ r, together with the relations ϕ_{i}(γ) < γ for γ ∈ F_{i} and 0 < i ≤ r.

We prove that F_{j−1} ∪ ⋯ ∪ F_{r} and P (ϕ_{r}, …, ϕ_{j}) = F_{r} ⊔_{ϕ_{r}} ⋯ ⊔_{ϕ_{j}} F_{j−1} agree as partially ordered subsets of C, by a descending induction on j. This is clear for j = r + 1, when we interpret both expressions as F_{r}. Thereafter, for 1 ≤ j ≤ r, we can inductively assume that F_{j} ∪ ⋯ ∪ F_{r} = P (ϕ_{r}, …, ϕ_{j+1}) as partially ordered subsets of C.

The order-preserving function ψ_{j}: P (ϕ_{r}, …, ϕ_{j+1}) → F_{j−1} is then equal to the function F_{j} ∪ ⋯ ∪ F_{r} → F_{j−1} given by γ ↦ γ ∩ L_{j−1}. The partial ordering on P (ϕ_{r}, …, ϕ_{j}) = P (ψ_{j}) agrees with the subset partial ordering when restricted to F_{j} ∪ ⋯ ∪ F_{r} = P (ϕ_{r}, …, ϕ_{j+1}), and when restricted to F_{j−1}. It remains to check that, for γ ∈ F_{j} ∪ ⋯ ∪ F_{r} and β ∈ F_{j−1}, we have β ≤ γ in the subset partial ordering on F_{j−1} ∪ (F_{j} ∪ ⋯ ∪ F_{r}) if and only if β ≤ γ in P (ψ_{j}). But this is clear, since β ≤ γ in P (ψ_{j}) if and only if β ≤ ψ_{j}(γ) in F_{j−1}, and ψ_{j}(γ) = γ ∩ L_{j−1} is the greatest face of γ that lies in F_{j−1}. □

Definition 4.12 ([9, 2.1.9]). A map f: A → B of finite simplicial sets is simple over U, for a given subset U ⊆ |B|, if for each b ∈ U the preimage |f|^{−1}(b) is contractible.

To summarize what we have learned so far, let κ be as in Proposition 1.3, let (z, w) be as in Notation 4.2, and let π: P (ϕ_{r}, …, ϕ_{1}) → [r] be as in Notation 4.8 and Definition 4.10.

Lemma 4.13. The map κ is simple over the open r-cell corresponding to (z, w) in the CW structure of | Sd Δ[m] × Sd Δ[n]|, if and only if Nπ is simple over the interior of |Δ[r]|.

5. Mapping cylinders

The (backward) reduced mapping cylinder, as introduced in [9, §2.4], is a functor that to each simplicial map f: X → Y associates a simplicial set M(f), factoring f as an inclusion in: X → M(f) followed by a simple projection map pr: M(f) → Y. There is a natural reduction map red: T (f) → M(f) from the ordinary mapping cylinder T (f) = X × Δ[1] ∪_{X} Y to the reduced mapping cylinder, and the ordinary coordinate projection T (f) → Δ[1] factors naturally through a reduced coordinate projection π: M(f) → Δ[1]:

[figure: X; in; f; T (f); red; M(f); pr; ≃_{s}; π; Y; Δ[1] .]

More generally, given a sequence X_{r} ⟶^{f_{r}} … ⟶^{f_{1}} X_{0} of maps of simplicial sets, there is an r-fold iterated reduced mapping cylinder M(f_{r}, …, f_{1}), a reduction map red: T (f_{r}, …, f_{1}) → M(f_{r}, …, f_{1}) from the r-fold iterated ordinary mapping cylinder, and compatible coordinate projection maps:

\xymatrix{T (f_{r}, …, f_{1}) \ar[r]^{red} \ar[dr] & M(f_{r}, …, f_{1}) \ar[d]^{π} \\  & Δ[r] .}

In this section, we shall recognize the map Nπ: N(P (ϕ_{r}, …, ϕ_{1})) → Δ[r] as the reduced coordinate projection map π: M(f_{r}, …, f_{1}) → Δ[r] for the sequence of simplicial sets X_{i} = NF_{i} and maps f_{i} = Nϕ_{i}. Thereafter, we use the second author’s criterion [9, 2.4.16] to show that the iterated reduction map T (f_{r}, …, f_{1}) → M(f_{r}, …, f_{1}) is simple in this case. This reduces the problem of showing that π is simple to the problem of showing that the classifying spaces |NF_{i}| are contractible, which we will address in the following, final, section.

Definition 5.1 ([9, 2.4.5]). Given an order-preserving function φ: V → W of partially ordered sets, let in: V → P (φ) = V ⊔_{φ} W and in′: W → P (φ) be the front and back inclusions, respectively, and let pr = φ∨1: P (φ) → W and π: P (φ) → [1] be as before. Consider the commutative diagram

\xymatrix{ & V \ar[dl]_{i_{1}} \ar[dr]^{in} \ar[drr]^{φ} \\ V × [1] \ar[rr]^{ρ} &  & P (φ) \ar[r]^{pr} \ar[d]^{π} & W \\ V \ar[u]^{i_{0}} \ar[r]^{φ} & W \ar[ur]^{_{in_{′}}} & [1]}

of order-preserving functions, where i_{t}(v) = (v, t), ρ(v, 1) = in(v) and ρ(v, 0) = in′(φ(v)). Applying nerves, we get the commutative diagram

\xymatrix{ & NV \ar[dl]_{i_{1}} \ar[d] \ar[dr]^{in} \ar[drr]^{Nφ} \\ NV × Δ[1] \ar[r] & T (Nφ) \ar[r]^{red} & N(P (φ)) \ar[r]^{pr} \ar[d]^{π} & NW \\ NV \ar[u]^{i_{0}} \ar[r]^{Nφ} & NW \ar[u] \ar[ur]^{_{in_{′}}} & Δ[1] ,}

where the lower left hand square is the pushout defining T (Nφ) = NV × Δ[1] ∪_{NV} NW, and the reduction map red is induced by Nρ and in′. We define the reduced mapping cylinder of the simplicial map f = Nφ: NV → NW to be the nerve M(f) = N(P (φ)) of the partially ordered set P (φ) = V ⊔_{ϕ} W, with inclusion, projection, reduction and coordinate projection maps as in the diagram above.

Example 5.2. Let [2] = {0 < 1 < 2} and [1]′ = {0′ < 1′}. The reduced mapping cylinders of the nerves of the elementary degeneracy operators σ_{0} : [2] → [1]′; 0, 1 ↦ 0′ (left) and σ_{1} : [2] → [1]; 1, 2 ↦ 1′ (right) are illustrated in the pictures below.

The extra relation 1′ < 1 in P (σ_{1}) = [2] ⊔_{σ_{1}} [1]′, which does not hold in P (σ_{0}) = [2] ⊔_{σ_{0}} [1]′, leads to the additional 3-simplex 0′ < 1′ < 1 < 2 on the right hand side.

The reduced mapping cylinder M(f) for a general simplicial map f: X → Y can be defined as a left Kan extension from these special cases, essentially as in [9, 2.4.4], but the definition just given suffices for our purposes. See [9, 2.4.12] for the compatibility of the two definitions.

We have already alluded to the following result.

Lemma 5.3. Let f: X → Y be any map of finite simplicial sets. The projection pr: M(f) → Y is a simple map.

Proof. See [9, 2.4.8]. □

The process of making reduced mapping cylinders can be iterated.

Definition 5.4. Suppose given a sequence V_{r} ⟶^{φ_{r}} … ⟶^{φ_{1}} V_{0} of partially ordered sets and order-preserving functions. Define the iterated reduced mapping cylinder of the sequence

X_{r} ⟶^{f_{r}} … ⟶^{f_{1}} X_{0}

of simplicial sets and maps, with X_{i} = NV_{i} and f_{i} = Nφ_{i}, to be the nerve

M(f_{r}, …, f_{1}) = N(P (φ_{r}, …, φ_{1}))

of the partially ordered set P (φ_{r}, …, φ_{1}) = V_{r} ⊔_{φ_{r}} ⋯ ⊔_{φ_{1}} V_{0}. Let the reduced coordinate projection π: M(f_{r}, …, f_{1}) → Δ[r] be the nerve of the order-preserving function π: P (φ_{r}, …, φ_{1}) → [r] that takes V_{i} to i, for each 0 ≤ i ≤ r.

The iterated reduced mapping cylinder M(f_{r}, …, f_{1}) of a general sequence X_{r} → ⋯ → X_{0} of simplicial sets and maps can, again, be defined as a left Kan extension from these special cases, but the definition given will suffice for this paper.

Lemma 5.5. The iterated reduced mapping cylinder M(f_{r}, …, f_{1}) is naturally isomorphic to the reduced mapping cylinder of the composite map

M(f_{r}, …, f_{2}) ⟶^{pr} X_{1} ⟶^{f_{1}} X_{0} ,

where the iterated projection pr: M(f_{r}, …, f_{1}) → X_{0} corresponds to the projection pr: M(f_{1} ∘ pr) → X_{0}.

Proof. Recall the notation from Definition 4.10. The composite of the two orderpreserving functions

P (φ_{r}, …, φ_{2}) ⟶^{ψ_{2}∨1} V_{1} ⟶^{φ_{1}} V_{0} ,

equals ψ_{1}. Hence the composite map f_{1} ∘ pr equals Nψ_{1}, and M(f_{r}, …, f_{1}) = N(P (φ_{r}, …, φ_{1})) = N(P (ψ_{1})) equals M(Nψ_{1}) = M(f_{1} ∘ pr), as asserted. □

Definition 5.6. Let X_{r} ⟶^{f_{r}} … ⟶^{f_{1}} X_{0} be any sequence of simplicial sets and maps. Define the iterated ordinary mapping cylinder T (f_{r}, …, f_{1}) = T (f_{1} ∘ pr) to be the ordinary mapping cylinder of the composite map

T (f_{r}, …, f_{2}) ⟶^{pr} X_{1} ⟶^{f_{1}} X_{0} ,

and let the iterated projection pr: T (f_{r}, …, f_{1}) → X_{0} be the ordinary projection pr: T (f_{1} ∘ pr) → X_{0}.

Definition 5.7. Suppose given any sequence V_{r} ⟶^{φ_{r}} … ⟶^{φ_{1}} V_{0} of partially ordered sets and order-preserving functions. Let the r-fold iterated reduction map

red: T (f_{r}, …, f_{1}) → M(f_{r}, …, f_{1})

be defined as the composite map

T (T (f_{r}, …, f_{2}) → X_{0}) → T (M(f_{r}, …, f_{2}) → X_{0}) ⟶^{red} M(M(f_{r}, …, f_{2}) → X_{0}) .

Here the left hand map is induced by the (r − 1)-fold iterated reduction map red: T (f_{r}, …, f_{2}) → M(f_{r}, …, f_{2}) and the identity map of X_{0}, while the right hand map is the reduction map for f_{1} ∘ pr: M(f_{r}, …, f_{2}) → X_{0}.

Notation 5.8. Consider the terminal sequence [0] → ⋯ → [0] of partially ordered sets and order-preserving functions. The order-preserving function π: [0]⊔_{φ_{r}} ⋯⊔_{φ_{1}} [0] → [r] is a bijection, so the r-fold iterated reduced mapping cylinder

M^{r} = M(Δ[0] → ⋯ → Δ[0]) = N([0] ⊔_{φ_{r}} ⋯ ⊔_{φ_{1}} [0])

maps isomorphically to Δ[r] under the reduced coordinate projection. Let T^{r} = T (Δ[0] → ⋯ → Δ[0]) be the r-fold iterated ordinary mapping cylinder. Then T^{0} ≅ Δ[0], T^{1} ≅ Δ[1], and T^{r} ≅ T^{r−1} × Δ[1] ∪_{T^{r−1}} Δ[0] for r ≥ 2. The iterated reduction map red: T^{r} → M^{r} agrees, up to the isomorphism π, with the ordinary cylinder projection T^{r} → Δ[r].

The map T^{r} → M^{r} is an isomorphism for r = 0 and 1. To address the case r ≥ 2, the following terminology will be convenient.

Definition 5.9 ([9, 2.4.7]). A map f: A → B of finite simplicial sets is a homotopy equivalence over the target if it has a section s: B → A such that |sf| is homotopic to the identity map on |A|, by a homotopy over |B|. The homotopy provides a contraction of each point inverse of |f|, so such a map f is simple.

Lemma 5.10. The iterated reduction map red: T^{r} → M^{r} is simple.

Proof. By induction, we may assume that r ≥ 2 and that T^{r−1} → M^{r−1} is simple. We must prove that the composite

T (T^{r−1} → Δ[0]) → T (M^{r−1} → Δ[0]) ⟶^{red} M(M^{r−1} → Δ[0])

is simple. The left hand map is simple by the inductive hypothesis and the gluing lemma for simple maps [9, 2.1.3(d)]. The right hand map is isomorphic to the reduction map for the unique map Nϵ: Δ[r − 1] → Δ[0], which is the canonical map from the pushout to the lower right hand entry in the commutative square

\xymatrix{Δ[r − 1] \ar[r]^{Nϵ} \ar[d]_{i_{0}} & Δ[0] \ar[d]^{_{in_{′}}} \\ Δ[r − 1] × Δ[1] \ar[r]^{Nρ} & N([r − 1] ⊔_{ϵ} [0]) .}

By the gluing lemma, and the fact that Nϵ is simple, it suffices to prove that Nρ is simple. There is an order-preserving bijection [r − 1] ⊔_{ϵ} [0] ≅ [r], taking 0 ∈ [0] to 0 ∈ [r] and i ∈ [r − 1] to i + 1 ∈ [r]. Under this identification, Nρ corresponds to Nθ, where θ: [r − 1] × [1] → [r] is given by θ(i, 0) = 0 and θ(i, 1) = i + 1.

To prove that Nθ: Δ[r − 1] × Δ[1] → Δ[r] is simple, it suffices to show that it is a homotopy equivalence over the target. The order-preserving function σ: [r] → [r − 1] × [1], given by σ(0) = (0, 0) and σ(j) = (j − 1, 1) for 1 ≤ j ≤ r, is a section of θ, and (σ ∘ θ)(i, t) ≤ (i, t) for all (i, t) ∈ [r − 1] × [1]. Hence Nσ is a section of Nθ, and there is a simplicial homotopy from Nσ ∘ Nθ to the identity, covering the identity of Δ[r]. □

Remark 5.11. Naturality of the iterated reduction map, with respect to the terminal morphism

\xymatrix{X_{r} \ar[r]^{f_{r}} \ar[d] & … \ar[r]^{f_{1}} & X_{0} \ar[d] \\ Δ[0] \ar[r] & … \ar[r] & Δ[0]}

in the category of r-tuples of composable maps, leads to the commutative diagram

\xymatrix{T (f_{r}, …, f_{1}) \ar[r]^{red} \ar[d] & M(f_{r}, …, f_{1}) \ar[d] \ar[dr]^{π} \\ T^{r} \ar[r]^{red}_{≃_{s}} & M^{r} \ar[r]_{≅} & Δ[r] .}

The lower reduction map T^{r} → M^{r} is simple, by the previous lemma. Each point inverse of the geometric realization |T (f_{r}, …, f_{1})| → |T^{r}| of the left hand vertical map is homeomorphic to one of the spaces |X_{0}|, …, |X_{r}|. This is obvious for r = 0 and 1, and follows by an easy induction for r ≥ 2. Hence the map T (f_{r}, …, f_{1}) → T^{r} is simple if (and only if) these spaces are contractible.

The left hand picture in Example 5.2 illustrates that the point inverses of the map |π|: |M(f_{r}, …, f_{1})| → |Δ[r]| need not be homeomorphic to one of the spaces |X_{0}|, …, |X_{r}|. This is why we introduce ordinary mapping cylinders and reduction maps, as technical tools.

In the remainder of this section we shall prove that the upper reduction map T (f_{r}, …, f_{1}) → M(f_{r}, …, f_{1}) is simple in the case when X_{i} = NF_{i} and f_{i} = Nϕ_{i} are as in Notation 4.8. In the following section we shall prove that the classifying spaces |NF_{i}| = |X_{i}| are contractible in that case. By the diagram above, and the right cancellation property for simple maps, it will then follow that π: M(f_{r}, …, f_{1}) → Δ[r] is simple. In view of Lemma 4.13, this will complete the proof of our main theorem.

The reduction map red: T (f) → M(f) is not always simple, as the restriction f: ∂Δ[2] → Δ[1] of Nσ_{0} : Δ[2] → Δ[1] illustrates; see the left hand figure in Example 5.2. When f = Nφ: NV → NW is the nerve of an order-preserving function of partially ordered sets, we have a useful criterion explained in Proposition 5.14 below.

Definition 5.12. Let V be a partially ordered set. A subset I ⊆ V is called a left ideal if v ∈ I and u ≤ v in V implies u ∈ I. It is called a right ideal if v ∈ I and v ≤ w in V implies w ∈ I. We always give I the subset partial ordering.

Definition 5.13. Suppose given a partially ordered set V and an element v ∈ V. Let V/v denote the left ideal in V consisting of all elements u ∈ V with u ≤ v. Any order-preserving function φ: V → W restricts to an order-preserving function φ/v: V/v → W/φ(v).

Proposition 5.14. Let φ: V → W be an order-preserving function of finite partially ordered sets. Then the following conditions are equivalent:

(1) The nerve of φ/v: V/v → W/φ(v) is simple, for each element v ∈ V.

(2) The reduction map red: T (Nφ) → M(Nφ) is simple.

Proof. See [9, 2.4.16]. □

Lemma 5.15. Let φ: V → W and pr = φ ∨ 1: P (φ) = V ⊔_{φ} W → W be as above. The nerve of the order-preserving function pr /v: P (φ)/v → W/ pr(v) is simple, for each v ∈ P (φ).

Proof. If v ∈ W ⊆ P (φ) then P (φ)/v = W/v, pr(v) = v and pr /v: W/v → W/v is the identity, whose nerve is obviously simple.

Otherwise, if v ∈ V ⊆ P (φ) then P (φ)/v = (V ⊔_{φ} W)/v is equal to P (φ/v) = V/v ⊔_{φ/v} W/φ(v), and pr(v) = φ(v), so we can identify pr /v with the orderpreserving function φ/v ∨ 1: P (φ/v) → W/φ(v). Its nerve is the projection map from the reduced mapping cylinder of N(φ/v) to its target, which is simple by [9, 2.4.8], recalled above as Lemma 5.3. □

We now return to the context of the previous section, where (z_{0}, w_{0}) < ⋯ < (z_{r}, w_{r}) is a non-degenerate r-simplex in Sd Δ[m]×Sd Δ[n], L_{i} = im(z_{i})×im(w_{i}) ⊂ [m] × [n], F_{i} is the set of (z_{i}, w_{i})-paths that are (z_{j}, w_{j})-full for every 0 ≤ j < i, and ϕ_{i}: F_{i} → F_{i−1} maps γ to ϕ_{i}(γ) = γ ∩ L_{i−1}.

Definition 5.16. The bijective correspondence between the elements γ: [k] → [m]×[n] of C = (Δ[m]×Δ[n])^{#} and the non-empty, totally ordered subsets im(γ) ⊆ [m] × [n] lets us define two further operations:

First, there is an order-preserving function C/γ × C/γ → C/γ that takes two paths α, β ∈ C/γ to the path α ∪ β ∈ C/γ with im(α ∪ β) = im(α) ∪ im(β).

Second, there is an order-preserving function F_{i} → C taking β ∈ F_{i} to the path β \ L_{i−1} with image im(β) \ L_{i−1}.

Lemma 5.17. The nerve of the order-preserving function

ϕ_{i}/γ: F_{i}/γ → F_{i−1}/ϕ_{i}(γ)

is simple, for each γ ∈ F_{i}. Therefore the reduction map T (Nϕ_{i}) → M(Nϕ_{i}) is simple, for each 1 ≤ i ≤ r.

Proof. We prove that N(ϕ_{i}/γ) is a homotopy equivalence over the target, in the sense of Definition 5.9. Using the operations mentioned above we can define an order-preserving function

σ: F_{i−1}/ϕ_{i}(γ) → F_{i}/γ

by α ↦ α ∪ (γ \ L_{i−1}). It is straightforward to check that σ is a section of ϕ_{i}/γ. If β ∈ F_{i}/γ, we see that

β = ϕ_{i}(β) ∪ (β \ L_{i−1}) ≤ ϕ_{i}(β) ∪ (γ \ L_{i−1}) = (σ ∘ ϕ_{i}/γ)(β) .

Hence there is a simplicial homotopy from the identity of N(F_{i}/γ) to Nσ∘N(ϕ_{i}/γ), covering the identity of N(F_{i−1}/ϕ_{i}(γ)).

The second assertion now follows from Proposition 5.14. □

Lemma 5.18. The r-fold iterated reduction map

red: T (Nϕ_{r}, …, Nϕ_{1}) → M(Nϕ_{r}, …, Nϕ_{1})

is simple, for each r ≥ 0.

Proof. This is trivial for r = 0, and was proved in Lemma 5.17 for r = 1. We may therefore suppose r ≥ 2, and assume, for some 1 ≤ i ≤ r, that the (r − i)-fold iterated reduction map red: T (Nϕ_{r}, …, Nϕ_{i+1}) → M(Nϕ_{r}, …, Nϕ_{i+1}) is simple. We must prove that the (r−i+1)-fold iterated reduction map red: T (Nϕ_{r}, …, Nϕ_{i}) → M(Nϕ_{r}, …, Nϕ_{i}) is simple. By definition, this is the composite of two further maps, each of which will be shown to be simple.

The first of the two is the map of ordinary mapping cylinders

T (T (Nϕ_{r}, …, Nϕ_{i+1}) → NF_{i−1}) → T (M(Nϕ_{r}, …, Nϕ_{i+1}) → NF_{i−1})

induced by the (r − i)-fold iterated reduction map and the identity on NF_{i−1}. It is simple by the inductive hypothesis and the gluing lemma for simple maps.

The second of the two is the reduction map

T (M(Nϕ_{r}, …, Nϕ_{i+1}) → NF_{i−1}) → M(M(Nϕ_{r}, …, Nϕ_{i+1}) → NF_{i−1})

associated to Nϕ_{i} ∘ pr: M(Nϕ_{r}, …, Nϕ_{i+1}) → NF_{i−1}. The latter map equals the nerve of the composite order-preserving function

P (ϕ_{r}, …, ϕ_{i+1}) ⟶^{ψ_{i+1}∨1} F_{i} ⟶^{ϕ_{i}} F_{i−1} ,

which we denote as ψ_{i} = ϕ_{i} ∘ (ψ_{i+1} ∨ 1). Recall from the proof of Lemma 5.5 that P (ϕ_{r}, …, ϕ_{i+1}) = P (ψ_{i+1}), where ψ_{i+1} : F_{r} ⊔_{ϕ_{r}} ⋯ ⊔_{ϕ_{i+2}} F_{i+1} → F_{i} is given by the composite ϕ_{i+1}∘⋯∘ϕ_{j} on F_{j}, so that ψ_{i+1}∨1 = pr: P (ψ_{i+1}) → F_{i} and ψ_{i} = ϕ_{i}∘pr. We will use the criterion in Proposition 5.14 to show that red: T (Nψ_{i}) → M(Nψ_{i}) is simple.

Consider any γ ∈ P (ϕ_{r}, …, ϕ_{i+1}) = P (ψ_{i+1}). We must show that the nerve of ψ_{i}/γ is simple. This order-preserving function is the composite of pr /γ and ϕ_{i}/ pr(γ). The nerve of pr /γ: P (ψ_{i+1})/γ → F_{i}/ pr(γ) is simple by Lemma 5.15, for the order-preserving function ψ_{i+1} and the element γ ∈ P (ψ_{i+1}). The nerve of ϕ_{i}/ pr(γ) is simple by Lemma 5.17, for the element pr(γ) ∈ F_{i}. Hence the composite N(ϕ_{i}/ pr(γ)) ∘ N(pr /γ) is also simple, as we needed to prove. □

6. Contracting sets of paths

As in the previous sections, we consider the map of nerves κ: Sd(Δ[m] × Δ[n]) → Sd Δ[m] × Sd Δ[n] induced by the order-preserving function (pr_{1}^{#}, pr_{2}^{#}): (Δ[m] × Δ[n])^{#} → Δ[m]^{#} × Δ[n]^{#}. We consider an r-simplex (z, w) in the target of κ, represented by a chain (z_{0} ≤ ⋯ ≤ z_{r}, w_{0} ≤ ⋯ ≤ w_{r}), where the z_{i} and w_{i} are faces of Δ[m] and Δ[n], respectively. However, unlike in the previous sections, we no longer assume that (z, w) is non-degenerate. This added generality will be convenient for our inductive proofs.

Notation 6.1. For any r-simplex (z, w) = (z_{0} ≤ ⋯ ≤ z_{r}, w_{0} ≤ ⋯ ≤ w_{r}) in the nerve of the partially ordered set Δ[m]^{#} × Δ[n]^{#}, let P^{z,w} be the partially ordered set of (z_{r}, w_{r})-paths γ: [k] → [m] × [n] that are (z_{j}, w_{j})-full for each 0 ≤ j < r, partially ordered as a subset of (Δ[m] × Δ[n])^{#}.

When (z, w) is non-degenerate, P^{z,w} is equal to the partially ordered set F_{r} from Notation 4.6. More generally, for each 0 ≤ i ≤ r the partially ordered set F_{i} is equal to P^{z′,w′}, where (z′, w′) = (z_{0} ≤ ⋯ ≤ z_{i}, w_{0} ≤ ⋯ ≤ w_{i}) is the front i-face of (z, w). As explained in Remark 5.11, the task that remains is to prove that each classifying space |NF_{i}| is contractible. Hence it will suffice to prove that |NP^{z,w}| is contractible, for each simplex (z, w). We shall prove this by induction on the dimension r ≥ 0 of that simplex.

We begin with the case r = 0, when (z, w) = (z_{0}, w_{0}) = (μ, ν) for some faces μ ∈ Δ[m]^{#} and ν ∈ Δ[n]^{#}. The proof we present for the following proposition introduces some ideas that will reappear in the cases r ≥ 1.

Proposition 6.2. The classifying space |NP^{μ,ν}| is contractible, for each pair of faces μ ∈ Δ[m]^{#} and ν ∈ Δ[n]^{#}.

Proof. Let ι_{m}: [m] → [m] and ι_{n}: [n] → [n] be the identity morphisms. In this proof it will be convenient to refer to (ι_{m}, ι_{n})-paths as (m, n)-paths, and to write P^{m,n} = P^{ι_{m},ι_{n}} for the partially ordered set formed by these paths. It consists of the injective order-preserving functions γ: [k] → [m] × [n] such that pr_{1} ∘γ: [k] → [m] and pr_{2} ∘γ: [k] → [n] are surjective, and is partially ordered so that β ≤ γ if and only if im(β) ⊆ im(γ).

Represent the faces μ and ν as injective order-preserving functions μ: [m_{1}] → [m] and ν: [n_{1}] → [n], respectively. Then there is an order-preserving bijection P^{m_{1},n_{1}} ≅ P^{μ,ν}, taking γ: [k] → [m_{1}] × [n_{1}] to (μ × ν)γ: [k] → [m] × [n]. It will therefore suffice to prove that the classifying space of P^{m_{1},n_{1}} is contractible, for each m_{1}, n_{1} ≥ 0. Changing the notation, we will prove that each partially ordered set P^{m,n} of (m, n)-paths has contractible classifying space. This will be done by induction on m ≥ 0.

Note first that any (m, n)-path γ: [k] → [m]×[n] must pass through γ(0) = (0, 0) and γ(k) = (m, n), due to the assumption that pr_{1} ∘γ and pr_{2} ∘γ are surjective.

For m = 0, P^{0,n} consists of a single path [n] → [0] × [n], taking i to (0, i), so |NP^{0,n}| is a single point.

For m = 1, P^{1,n} consists of n “short” paths [n] → [1] × [n] (for each 0 ≤ j < n there is one such path mapping j and j + 1 to (0, j) and (1, j + 1), respectively), and n + 1 “long” paths [n + 1] → [1] × [n] (for each 0 ≤ j ≤ n there is one such path mapping j and j + 1 to (0, j) and (1, j), respectively). These are partially ordered as a zig-zag of length 2n, as illustrated below for n = 2.

Hence the nerve NP^{1,n} is a union of 2n edges, with alternating orientations, and the classifying space |NP^{1,n}| is homeomorphic to an interval.

To handle the general case, fix some m ≥ 1 and n ≥ 0, and suppose inductively that |NP^{m_{1},n_{1}}| is contractible whenever 0 ≤ m_{1} ≤ m and n_{1} ≥ 0. We shall prove that the classifying space of P^{m+1,n} is contractible.

For each 0 ≤ j ≤ n, let Q_{j} ⊂ P^{m+1,n} be the partially ordered subset consisting of the paths γ: [k] → [m+1]×[n] that pass through (m, j), i.e., so that (m, j) ∈ im(γ). Each (m + 1, n)-path γ has to pass through some (m, j), since pr_{2} ∘γ: [k] → [m + 1] is surjective, so the Q_{j} cover P^{m+1,n}. Furthermore, these are right ideals in P^{m+1,n}, so their nerves NQ_{j} also cover NP^{m+1,n}, and

|NP^{m+1,n}| = |NQ_{0}| ∪ ⋯ ∪ |NQ_{n}|

is a finite union of CW complexes. By a Mayer–Vietoris type argument, it will therefore suffice to prove that each ℓ-fold intersection

|NQ_{j_{1}}| ∩ ⋯ ∩ |NQ_{j_{ℓ}}| = |N(Q_{j_{1}} ∩ ⋯ ∩ Q_{j_{ℓ}})|

is contractible, for 0 ≤ j_{1} < ⋯ < j_{ℓ} ≤ n and ℓ ≥ 1. In fact, we only need to consider single and double intersections, since Q_{j_{1}} ∩ ⋯ ∩ Q_{j_{ℓ}} = Q_{j_{1}} ∩ Q_{j_{ℓ}} in our context.

In the first case, there is an order-preserving bijection

Q_{j} ≅ P^{m,j} × P^{1,n−j}

for each 0 ≤ j ≤ n, breaking an (m + 1, n)-path γ passing through (m, j) into two pieces: γ^{1} in [m] × [j] and γ^{2} in {m < m + 1} × {j < ⋯ < n} ≅ [1] × [n − j].

Hence |NQ_{j}| ≅ |NP^{m,j}| × |NP^{1,n−j}| is a product of two contractible spaces, by our inductive hypothesis.

In the second case, there is an order-preserving bijection

Q_{i} ∩ Q_{j} ≅ P^{m,i} × P^{0,j−i} × P^{1,n−j}

for each 0 ≤ i < j ≤ n, breaking a path γ passing through (m, i) and (m, j) into three pieces: γ^{1} in [m] × [i], γ^{2} in {m} × {i < ⋯ < j} ≅ [0] × [j − i] and γ^{3} in {m < m + 1} × {j < ⋯ < n} ≅ [1] × [n − j]. Hence |N(Q_{i} ∩ Q_{j})| ≅ |NP^{m,i}| × |NP^{0,j−i}| × |NP^{1,n−j}| is a product of three contractible spaces, by the inductive hypothesis. □

To handle the cases r ≥ 1, we will use the following notation.

Notation 6.3. For m, n ≥ 0, let C^{m,n} = (Δ[m] × Δ[n])^{#} be the partially ordered set of paths γ, previously denoted C, where γ: [k] → [m] × [n] is order-preserving and injective. Let D^{m,n} = Δ[m]^{#} × Δ[n]^{#} be the partially ordered set of pairs of faces (μ, ν), where μ: [m_{1}] → [m] and ν: [n_{1}] → [n] are order-preserving and injective. The order-preserving function (pr_{1}^{#}, pr_{2}^{#}): C^{m,n} → D^{m,n} induces the canonical map κ upon passage to nerves.

For any (p, q) ∈ [m] × [n], let C_{(p,q)}^{m,n} be the subset of paths γ ∈ C^{m,n} that pass through (p, q), so that γ(j) = (p, q) for some j ∈ [k]. Let D_{(p,q)}^{m,n} be the subset of pairs (μ, ν) ∈ D^{m,n} such that (p, q) ∈ im(μ)×im(ν). When (p, q) ∈ im(z_{0})×im(w_{0}), let P_{(p,q)}^{z,w} be the subset of (z, w)-paths γ ∈ P^{z,w} that pass through (p, q), in the same sense as above. Each of these subsets is given the subset partial ordering.

Lemma 6.4. (a) There is an order-preserving bijection

C_{(p,q)}^{m,n} ≅ C_{(p,q)}^{p,q} × C_{(0,0)}^{m−p,n−q}

taking γ: [k] → [m] × [n] with γ(j) = (p, q) to (γ^{1}, γ^{2}), where γ^{1} : [j] → [p] × [q] and γ^{2} : [k − j] → [m − p] × [n − q] are given by γ^{1}(i) = γ(i) for i ∈ [j] and γ^{2}(i) + (p, q) = γ(i + j) for i ∈ [k − j].

(b) There is an order-preserving bijection

D_{(p,q)}^{m,n} ≅ D_{(p,q)}^{p,q} × D_{(0,0)}^{m−p,n−q}

taking (μ, ν) to ((μ^{1}, ν^{1}), (μ^{2}, ν^{2})), where (if μ: [m_{1}] → [m] and μ(j) = p) μ^{1} : [j] → [p] and μ^{2} : [m_{1} − j] → [m − p] are given by μ^{1}(i) = μ(i) and μ^{2}(i) + p = μ(i + j), and similarly for ν, ν^{1} and ν^{2}.

(c) Suppose that (p, q) ∈ im(z_{0}) × im(w_{0}). The bijection in (a) restricts to an order-preserving bijection

P_{(p,q)}^{z,w} ≅ P^{z^{1},w^{1}} × P^{z^{2},w^{2}}

taking a (z, w)-path γ passing through (p, q) to (γ^{1}, γ^{2}), where γ^{1} is a (z^{1}, w^{1})- path and γ^{2} is a (z^{2}, w^{2})-path. Here the bijection in (b) takes z_{i} to (z_{i}^{1}, z_{i}^{2}) and w_{i} to (w_{i}^{1}, w_{i}^{2}) for each i, and (z^{1}, w^{1}) = (z_{0}^{1} ≤ ⋯ ≤ z_{r}^{1}, w_{0}^{1} ≤ ⋯ ≤ w_{r}^{1}) and (z^{2}, w^{2}) = (z_{0}^{2} ≤ ⋯ ≤ z_{r}^{2}, w_{0}^{2} ≤ ⋯ ≤ w_{r}^{2}).

Proof. Part (a) is clear, since any path γ: [k] → [m] × [n] with γ(j) = (p, q) must map [j] into [p] × [q] and {j < ⋯ < k} ≅ [k − j] into {p < ⋯ < m} × {q < ⋯ < n} ≅ [m − p] × [n − q].

Part (b) is also clear, since any face μ: [m_{1}] → [m] with μ(j) = p must map [j] into [p] and {j < ⋯ < k} ≅ [k − j] into {p < ⋯ < m} ≅ [m − p], and likewise for ν.

To prove (c), note that a path γ passing through (p, q) is a (z_{r}, w_{r})-path if and only if γ^{1} (passing through (p, q)) is a (z_{r}^{1}, w_{r}^{1})-path and γ^{2} (passing through (0, 0)) is a (z_{r}^{2}, w_{r}^{2})-path. The condition that γ^{1} passes through (p, q) is automatic, since (p, q) ∈ im(z_{0}^{1}) × im(w_{0}^{1}) ⊆ im(z_{r}^{1}) × im(w_{r}^{1}), so a (z_{r}^{1}, w_{r}^{1})-path in [p] × [q] must end at (p, q), and similarly for the condition that γ^{2} passes through (0, 0).

Finally, for each 0 ≤ j < r, the (z_{r}, w_{r})-path γ passing through (p, q) is (z_{j}, w_{j})- full if and only if γ^{1} is (z_{j}^{1}, w_{j}^{1})-full and γ^{2} is (z_{j}^{2}, w_{j}^{2})-full. This follows from the result in the previous paragraph, together with the observation that the path γ ∩ im(z_{j}) × im(w_{j}), with image im(γ) ∩ im(z_{j}) × im(w_{j}), passes through (p, q), and corresponds under the bijection in (a) with the pair of paths γ^{1} ∩ im(z_{j}^{1}) × im(w_{j}^{1}) and γ^{2} ∩ im(z_{j}^{2}) × im(w_{j}^{2}). □

Notation 6.5. For (p, q) ≤ (p′, q′) in [m] × [n], let C_{(p,q),(p′,q′)}^{m,n} = C_{(p,q)}^{m,n} ∩ C_{(p′,q′)}^{m,n} be the set of paths that pass through both (p, q) and (p′, q′). Let D_{(p,q),(p′,q′)}^{m,n} = D_{(p,q)}^{m,n}∩ D_{(p′,q′)}^{m,n} be the set of pairs (μ, ν) such that (p, q) and (p′, q′) lie in im(μ) × im(ν). When (p, q), (p′, q′) ∈ im(z_{0}) × im(w_{0}), let P_{(p,q),(p′,q′)}^{z,w} = P_{(p,q)}^{z,w} ∩ P_{(p′,q′)}^{z,w} be the set of (z, w)-paths that pass through both (p, q) and (p′, q′).

Lemma 6.6. Suppose that (p, q) ≤ (p′, q′) in im(z_{0}) × im(w_{0}).

(a) There is an order-preserving bijection

D_{(p,q),(p′,q′)}^{m,n} ≅ D_{(p,q)}^{p,q} × D_{(0,0),(p′−p,q′−q)}^{p′−p,q′−q} × D_{(0,0)}^{m−p′,n−q′}

taking (μ, ν) to ((μ^{1}, ν^{1}), (μ^{2}, ν^{2}), (μ^{3}, ν^{3})).

(b) There is an order-preserving bijection

P_{(p,q),(p′,q′)}^{z,w} ≅ P^{z^{1},w^{1}} × P^{z^{2},w^{2}} × P^{z^{3},w^{3}}

taking the (z, w)-paths γ that pass through both (p, q) and (p′, q′), to the triples (γ^{1}, γ^{2}, γ^{3}), where γ^{ℓ} is a (z^{ℓ}, w^{ℓ})-path for each 1 ≤ ℓ ≤ 3.

Proof. This follows by two applications of the previous lemma, or by a direct proof. In (b), (z^{ℓ}, w^{ℓ}) = (z_{0}^{ℓ} ≤ ⋯ ≤ z_{r}^{ℓ}, w_{0}^{ℓ} ≤ ⋯ ≤ w_{r}^{ℓ}) for each 1 ≤ ℓ ≤ 3, where z_{i} corresponds to (z_{i}^{1}, z_{i}^{2}, z_{i}^{3}) as in (a), and likewise for w_{i}. □

We can now prove our main technical result.

Proposition 6.7. The classifying space |NP^{z,w}| is contractible, for each simplex (z, w) in Sd Δ[m] × Sd Δ[n].

Proof. The proof involves a double induction. We begin with an outer induction on the dimension r ≥ 0 of the simplex (z, w). The initial case r = 0 was handled in Proposition 6.2. For the outer inductive step, we fix an r ≥ 0 and assume that |NP^{z,w}| is contractible for each r-simplex (z, w). We shall prove that |NP^{x,y}| is contractible for each (r + 1)-simplex

(x, y) = (x_{0} ≤ ⋯ ≤ x_{r+1}, y_{0} ≤ ⋯ ≤ y_{r+1})

in the nerve of Δ[m]^{#} × Δ[n]^{#}.

Let the pair (ζ, η) be the 0-th vertex of (x, y), and let the r-simplex (z, w) be the 0-th face of (x, y). In other words, let ζ = x_{0} : [s] → [m] and η = y_{0} : [t] → [n], for some s, t ≥ 0, and let z_{i} = x_{i+1} and w_{i} = y_{i+1} for all 0 ≤ i ≤ r. With this notation, an (x, y)-path is the same as a (z, w)-path that is (ζ, η)-full. We maintain this notational scheme throughout this proof.

We shall prove that the classifying space of P^{x,y} is contractible, by an inner induction on the dimension s ≥ 0 of the face ζ of Δ[m].

The inner induction begins with the case s = 0: When ζ: [0] → [m] is 0dimensional, an (x, y)-path is the same as a (z, w)-path that goes through both (ζ(0), η(0)) and (ζ(0), η(t)). Hence

P^{x,y} = P_{(ζ(0),η(0)),(ζ(0),η(t))}^{z,w} ≅ P^{z^{1},w^{1}} × P^{z^{2},w^{2}} × P^{z^{3},w^{3}}

by Lemma 6.6. Here (z_{i}^{1}, w_{i}^{1}) ∈ D^{ζ(0),η(0)}, (z_{i}^{2}, w_{i}^{2}) ∈ D^{0,η(t)−η(0)} and (z_{i}^{3}, w_{i}^{3}) ∈ D^{m−ζ(0),n−η(t)}, for all 0 ≤ i ≤ r. Each (z^{ℓ}, w^{ℓ}) is an r-simplex, for 1 ≤ ℓ ≤ 3, so |NP^{x,y}| is a product of three contractible spaces by the outer inductive hypothesis, and is therefore contractible.

The inner induction continues with the case s = 1: When ζ: [1] → [m] is 1-dimensional, any (ζ, η)-full path must go through (ζ(0), η(0)) and (ζ(1), η(t)). Hence

P^{x,y} ≅ P^{x^{1},y^{1}} × P^{x^{2},y^{2}} × P^{x^{3},y^{3}}

by Lemma 6.6. Here (x_{i}^{1}, y_{i}^{1}) ∈ D^{ζ(0),η(0)}, (x_{i}^{2}, y_{i}^{2}) ∈ D^{ζ(1)−ζ(0),η(t)−η(0)} and (x_{i}^{3}, y_{i}^{3}) ∈ D^{m−ζ(1),n−η(t)}, for all 0 ≤ i ≤ r + 1.

The part ζ^{1} = x_{0}^{1} of ζ: [1] → [n] that lands in [ζ(0)] is 0-dimensional, and similarly for the part ζ^{3} = x_{0}^{3}. Hence P^{x^{1},y^{1}} and P^{x^{3},y^{3}} have contractible classifying spaces, by the already established case s = 0 of the inner induction. It therefore remains to prove that P^{x^{2},y^{2}} has contractible classifying space. Replacing (x^{2}, y^{2}) with (x, y) in the notation, we may and will assume that ζ(0) = 0, ζ(1) = m, η(0) = 0 and η(t) = n, and seek to prove that P^{x,y} has contractible classifying space.

By symmetry, the case t = 0 can be handled just like the case s = 0. We therefore assume t ≥ 1. For each 0 ≤ j ≤ t − 1 let

Q_{j} = P_{(0,η(j)),(m,η(j+1))}^{x,y}

be the subset of P^{x,y} of paths that go through both (0, η(j)) and (m, η(j + 1)), where ζ(0) = 0 and ζ(1) = m. Each (ζ, η)-full path must go through these two points for some 0 ≤ j ≤ t − 1, so P^{x,y} = Q_{0} ∪ ⋯ ∪ Q_{t−1}. Each Q_{j} is a right ideal, so |NP^{x,y}| = |NQ_{0}| ∪ ⋯ ∪ |NQ_{t−1}|.

We now argue that each |NQ_{j}| is contractible. Since every path in Q_{j} goes through both (0, η(j)) and (m, η(j + 1)), we have an order-preserving bijection

Q_{j} ≅ P^{x^{1},y^{1}} × P^{x^{2},y^{2}} × P^{x^{3},y^{3}}

with (x_{i}^{1}, y_{i}^{1}) ∈ D^{0,η(j)}, (x_{i}^{2}, y_{i}^{2}) ∈ D^{m,η(j+1)−η(j)} and (x_{i}^{3}, y_{i}^{3}) ∈ D^{0,n−η(j+1)}. Once more, ζ^{1} = x_{0}^{1} and ζ^{3} = x_{0}^{3} are 0-dimensional, so P^{x^{1},y^{1}} and P^{x^{3},y^{3}} have contractible classifying spaces by the case s = 0. On the other hand, ζ^{2} = x_{0}^{2} = ζ and η^{2} = y_{0}^{2} : [1] → [η(j + 1) − η(j)] are both 1-dimensional.

We claim that the classifying space of P^{x^{2},y^{2}} is also contractible. Since both ζ^{2} and η^{2} are 1-dimensional, a (z^{2}, w^{2})-path is (ζ^{2}, η^{2})-full if and only if it goes through both (ζ^{2}(0), η^{2}(0)) = (0, 0) and (ζ^{2}(1), η^{2}(1)) = (m, η(j + 1) − η(j)). By assumption, ζ^{2} = x_{0}^{2} ≤ x_{1}^{2} = z_{0}^{2} and η^{2} = y_{0}^{2} ≤ y_{1}^{2} = w_{0}^{2}, so any (z_{0}^{2}, w_{0}^{2})-full path in [m] × [η(j + 1) − η(j)] will begin at (0, 0) and end at (m, η(j + 1) − η(j)). Hence

P^{x^{2},y^{2}} = P_{(0,0),(m,η(j+1)−η(j))}^{z^{2},w^{2}} = P^{z^{2},w^{2}} ,

which has contractible classifying space by the outer inductive hypothesis, since (z^{2}, w^{2}) is an r-simplex. This completes the proof that each |NQ_{j}| is contractible.

Next we consider the Q_{i} ∩ Q_{j} for 0 ≤ i < j ≤ t − 1. If i + 1 < j there are no paths that go through both (ζ(1), η(i + 1)) and (ζ(0), η(j)), so in these cases Q_{i} ∩Q_{j} = ∅. This implies that all ℓ-fold intersections Q_{j_{1}} ∩⋯∩Q_{j_{ℓ}} = ∅ are empty, for 0 ≤ j_{1} < ⋯ < j_{ℓ} ≤ t − 1 and ℓ ≥ 3.

It remains to consider the double intersection Q_{j−1} ∩ Q_{j}, for 1 ≤ j ≤ t − 1. It consists of the (x, y)-paths that go through the four points (0, η(j − 1)), (0, η(j)), (m, η(j)) and (m, η(j + 1)), where ζ(0) = 0 and ζ(1) = m. There is a unique such path, with image contained in the totally ordered subset

{0} × [η(j)] ∪ [m] × {η(j)} ∪ {m} × {η(j) < ⋯ < n}

of [m] × [n]. Hence |NQ_{j−1}| ∩ |NQ_{j}| is a single point.

It follows that the union |NQ_{0}|∪⋯∪|NQ_{t−1}| is homotopy equivalent to the set of points {j | 0 ≤ j ≤ t − 1}, connected by the intervals [j − 1, j] for 1 ≤ j ≤ t − 1. Their union is the interval [0, t − 1], which is contractible. This completes the proof for s = 1 and t ≥ 1.

The inner induction ends with the inner inductive step, for s ≥ 2: Assume inductively that |NP^{x′,y′}| is contractible for each (r + 1)-simplex (x′, y′) such that the dimension of ζ′ = x′_{0} is strictly less than s. We must prove that P^{x,y} has contractible classifying space when the dimension of ζ = x_{0}, as a face of Δ[m], equals s.

For each 0 ≤ j ≤ t, where t is the dimension of η = y_{0}, let

Q_{j} = P_{(ζ(s−1),η(j))}^{x,y}

be the partially ordered set of (x, y)-paths that go through (ζ(s − 1), η(j)). Each (x, y)-path is (ζ, η)-full, hence must pass through one of these points, so P^{x,y} = Q_{0} ∪ ⋯ ∪ Q_{t}. Each Q_{j} is a right ideal, so |NP^{x,y}| = |NQ_{0}| ∪ ⋯ ∪ |NQ_{t}| is a finite union of CW complexes.

By Lemma 6.4 there is an order-preserving bijection

Q_{j} ≅ P^{x^{1},y^{1}} × P^{x^{2},y^{2}}

where (x^{1}, y^{1}) and (x^{2}, y^{2}) are (r + 1)-simplices, with ζ^{1} = x_{0}^{1} of dimension s − 1 and ζ^{2} = x_{0}^{2} of dimension 1. Hence |NP^{x^{1},y^{1}}| and |NP^{x^{2},y^{2}}| are contractible by the inner inductive hypothesis, and this implies that |NQ_{j}| is contractible.

For 0 ≤ i < j ≤ t, the double intersection

Q_{i} ∩ Q_{j} = P_{(ζ(s−1),η(i)),(ζ(s−1),η(j))}^{x,y}

consists of the (x, y)-paths that pass through (ζ(s − 1), η(i)) and (ζ(s − 1), η(j)). By Lemma 6.6 there is an isomorphism

Q_{i} ∩ Q_{j} ≅ P^{x^{1},y^{1}} × P^{x^{2},y^{2}} × P^{x^{3},y^{3}}

of partially ordered sets, where ζ^{1} = x_{0}^{1} is of dimension s − 1, ζ^{2} = x_{0}^{2} is of dimension 0, and ζ^{3} = x_{0}^{3} is of dimension 1. Each of the three factors has contractible classifying space, by the inner inductive hypothesis, so |NQ_{i}| ∩ |NQ_{j}| is also contractible.

Finally, for any 0 ≤ j_{1} < ⋯ < j_{ℓ} ≤ t with ℓ ≥ 3, the ℓ-fold intersection

Q_{j_{1}} ∩ ⋯ ∩ Q_{j_{ℓ}} = Q_{j_{1}} ∩ Q_{j_{ℓ}}

equals one of the double intersections considered above. This is because any (ζ, η)- full path that passes through (ζ(s−1), η(j_{1})) and (ζ(s−1), η(j_{ℓ})) must pass through (ζ(s − 1), η(i)) for each j_{1} ≤ i ≤ j_{ℓ}.

Hence we have proved that each finite intersection |NQ_{j_{1}}| ∩ ⋯ ∩ |NQ_{j_{ℓ}}| is contractible, which readily implies that the union |NP^{x,y}| = |NQ_{0}| ∪ ⋯ ∪ |NQ_{t}| is contractible. □

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Department of Mathematics, University of Oslo, Norway

E-mail address: rvfjellb@math.uio.no

Department of Mathematics, University of Oslo, Norway

E-mail address: rognes@math.uio.no
URL: http://folk.uio.no/rognes