PUBLICATIONS DE L’INSTITUT MATHÉMATIQUE Nouvelle série, tome 97(111) (2015), 11–21 DOI: 10.2298/PIM141125001P

SEGAL’S MULTISIMPLICIAL SPACES

Zoran Petrić

Abstract. Some sufficient conditions on a simplicial space X : Δ^{op} → Top guaranteeing that X_{1} ≃ Ω|X| were given by Segal. We give a generalization of this result for multisimplicial spaces. This generalization is appropriate for the reduced bar construction, providing an n-fold delooping of the classifying space of a category.

1. Introduction

This note makes no great claim to originality. It provides a complete inductive argument for a generalization of [17, Proposition 1.5], which was spelled out, not in a precise manner, in [2, paragraph preceding Theorem 2.1]. The authors of [2] did not provide the proof of this generalization. Some related, but quite different, results are given in [4] and [3].

The main result of [5] reaches its full potential role in constructing a model for an n-fold delooping of the classifying space of a category only with the help of such a generalization of [17, Proposition 1.5]. The aim of this note is to fill in a gap in the literature concerning these matters.

Segal, [17, Proposition 1.5], gave conditions on a simplicial space X : Δ^{op} → Top guaranteeing that X_{1} ≃ Ω|X|. His intention was to cover a more general class of simplicial spaces than we need for our purposes, therefore he worked with nonstandard geometric realizations of simplicial spaces. We generalize his result, in one direction, by passing from simplicial spaces to multisimplicial spaces, but staying in a class appropriate for the standard geometric realization. Our result is formulated to be directly applicable to the reduced bar construction of [5], providing an n-fold delooping of the classifying space of a category.

We work in the category (here denoted by Top) of compactly generated Hausdorff spaces. (This category is denoted by Ke in [7] and by CGHaus in [9].) The objects of Top are called spaces and the arrows are called maps. Products are given the compactly generated topology. We adopt the following notation throughout: ≃ for homotopy of maps or same homotopy type of spaces, ≈ for homeomorphism of spaces.

2010 Mathematics Subject Classification: 18G30; 55P35; 55P48.

Key words and phrases: simplicial space, loop space, lax functor.

I am grateful to anonymous referees for some useful comments concerning the previous version of this note. This work was supported by project ON174026 of the Ministry of Education, Science, and Technological Development of the Republic of Serbia.

Communicated by Žarko Mijajlović.

The category Δ (denoted by Δ^{+} in [9]) is the standard topologist’s simplicial category defined as in [9, VII.5]. We identify this category with the subcategory of Top whose objects are the standard ordered simplices (one for each dimension), i.e., with the standard cosimplicial space Δ → Top.

The objects of Δ are the nonempty ordinals 1, 2, 3, …, which are rewritten as [0], [1], [2], etc. The coface arrows from [n − 1] to [n] are denoted by δ_{i}^{n}, for 0 ⩽ i ⩽ n, and the codegeneracy arrows from [n] to [n − 1] are denoted by σ_{i}^{n}, for 0 ⩽ i ⩽ n − 1.

For the opposite category Δ^{op}, we denote the arrow (δ_{i}^{n})^{op} : [n] → [n − 1] by d_{i}^{n} and (σ_{i}^{n})^{op} : [n − 1] → [n] by s_{i}^{n}. For f an arrow of Δ^{op} (or (Δ^{op})^{n}), we abbreviate X(f) by f whenever the (multi)simplicial object X is determined by the context.

We consider all the monoidal structures to be strict, which is supported by the strictification given by [9, XI.3, Theorem 1]. Some proofs prepared for nonspecialists are given in the appendix.

2. Multisimplicial spaces and their realization

A multisimplicial space is an object of the category Top^{(Δ^{op} )^{n}} , i.e., a functor from (Δ^{op})^{n} to Top. When n = 0, this is just a space and when n = 1, this is a simplicial space. As usual, for a multisimplicial space X : (Δ^{op})^{n} → Top, we abbreviate X([k_{1}], …, [k_{n}]) by X_{k_{1} ...k_{n}} .

We say that X : (Δ^{op})^{n} → Top is a multisimplicial set when every X_{k_{1} ...k_{n}} is discrete. A multisimplicial map is an arrow of Top^{(Δ^{op} )^{n}} , i.e., a natural transformation between multisimplicial spaces. When n = 1, this is a simplicial map. Throughout this paper we use the standard geometric realization of (multi)simplicial spaces.

Definition 2.1. The realization functor | | : Top^{Δ^{op}} → Top of simplicial spaces is defined so that for a simplicial space X, we have

|X| = ( ∐_{n} X_{n} × Δ^{n}) / ∼,

where the equivalence relation ∼ is generated by

(d_{i}^{n}x, t) ∼ (x, δ_{i}^{n}t) and (s_{i}^{n}x, t) ∼ (x, σ_{i}^{n}t).
Definition 2.2. For p ⩾ 0, the functor ^{(p)} : Top^{(Δ^{op} )^{n+p}} → Top^{(Δ^{op} )^{n}} of

partial realization is defined inductively as follows

(0) is the identity functor, and ^{(p+1)} is the composition
Top^{(Δ^{op} )^{n+p+1}}  ⟶^{≅} (Top^{Δ^{op}} )^{(Δ^{op} )^{n+p}}  −−−−−−−→^{| |^{(Δ^{op} )^{n+p}}} Top^{(Δ^{op} )^{n+p}}  −−→^{(p)} Top^{(Δ^{op} )^{n}} ,

where the first isomorphism maps X to Y such that (Y_{k_{1} ...k_{n+p}} )_{l} = X_{k_{1} ...k_{n+p} l}.

For a multisimplicial space X : (Δ^{op})^{p} → Top, we denote X^{(p)} by |X|. Hence,

for X : (Δ^{op})^{n+p} → Top, we have that (X^{(p)})_{k_{1} ...k_{n}} = |X_{k_{1} ...k_{n} \underbrace{…}_{p}}|.

Definition 2.3. If X = Y^{(p)}, for Y a multisimplicial set, then we say that X is a partially realized multisimplicial set (PRmss).

Definition 2.4. For n ⩾ 0 and X : (Δ^{op})^{n} → Top, let the simplicial space diagX : Δ^{op} → Top be such that

(diagX)_{k} = X_{k...k}.

In particular, when n = 0 and X is just a topological space, we have that (diagX)_{k} is X and all the face and degeneracy maps of diagX are 1_{X}.

The following lemma is a corollary of [15, Lemma, p. 94].

Lemma 2.5. For X : (Δ^{op})^{n} → Top, we have that |X| ≈ | diagX|.

As a consequence of Lemma 2.5 and [10, Theorem 14.1] we have the following lemma.

Lemma 2.6. If X is a PRmss, then |X| is a CW-complex.

The following remark easily follows.

Remark 2.7. (a) If X : (Δ^{op})^{n+p} → Top is a PRmss, then X^{(p)} is a PRmss.

(b) If X : (Δ^{op})^{n} → Top is a PRmss, then for every k_{1}, …, k_{n}, the space X_{k_{1} ...k_{n}} is a CW-complex.

(c) If X : (Δ^{op})^{n} → Top is a PRmss, then for every k ⩾ 0, X_{k ...} is a PRmss.

(d) If X : (Δ^{op})^{n} → Top, for n > 1, is a PRmss, then Y : Δ^{op} ×Δ^{op} → Top, defined so that Y_{mk} = X_{mk...k} , is a PRmss.

Definition 2.8. A simplicial space X : Δ^{op} → Top is good when for every 0 ⩽ i ⩽ n − 1, the map s_{i}^{n} : X_{n−1} → X_{n} is a closed cofibration. It is proper (Reedy cofibrant) when for every n ⩾ 1, the inclusion sX_{n} ↪ X_{n}, where sX_{n} = ⋃ _{i} s_{i}^{n}(X_{n−1} ), is a closed cofibration.

Proposition 2.9. Every PRmss X : Δ^{op} → Top is good.

Proof. Since d_{i}^{n} ∘ s_{i}^{n} = 1_{X_{n−1}} , we may consider X_{n−1} to be a retract of X_{n}. By Remark 2.7(b), X_{n} is a CW-complex and by [6, Corollary III.2] (see also [8, Corollary 2.4(a)]) a locally equiconnected space. By [8, Lemma 3.1] and since every X_{k} is Hausdorff, s_{i}^{n} is a closed cofibration. ⊣

As a corollary of [18, Proposition 22] (see also references listed in [18, Section 6, p. 19]) we have the following result.

Lemma 2.10. Every good simplicial space is proper.

The following result is from [12, Appendix, Theorem A4(ii)].

Lemma 2.11. Let f : X → Y be a simplicial map of proper simplicial spaces. If each f_{k}: X_{k} → Y_{k} is a homotopy equivalence, then |f|: |X| → |Y | is a homotopy equivalence.

Definition 2.12. The product X × Y of simplicial spaces X and Y is defined componentwise, i.e., (X × Y)_{k} = X_{k} × Y_{k}, and for every arrow f: k → l of Δ^{op} and every x ∈ X_{k} and y ∈ Y_{k}, we have that f(x, y) = (fx, fy).

Since the product of two CW-complexes in Top is a CW-complex, by reasoning as in Proposition 2.9, we have the following.

Remark 2.13. If X, Y : Δ^{op} → Top are PRmss, then X × Y is good.

The following lemma is a corollary of [11, Lemma 11.11].

Lemma 2.14. If the space X_{0} of a simplicial space X : Δ^{op} → Top is pathconnected, then |X| is path-connected.

3. Segal’s multisimplicial spaces

For m ⩾ 1, consider the arrows i_{1}, …, i_{m} : [m] → [1] of Δ^{op} given by the following diagrams.

[figure: i_{1} :; …; 0; 1; 0; 1; m; i_{2} :; …; 0; 1; 0; 1; 2; m; …; i_{m} :; …; 0; 1; 0; m−1; m]

The following images of these arrows under the functor J : Δ^{op} → Δ of [14, Section 6] may help the reader to see that i_{1}, …, i_{m} correspond to m projections. (Note that 0 and 2 in the bottom line of the images serve to project away all but one element of the top line.)

[figure: …; 0; 1; 2; 0; 1; 2; m; m+1; …; 0; 1; 2; 0; 1; 2; 3; m+1; …; …; 0; 1; 2; 0; 1; m−1; m; m+1]
For maps f_{i}: A → B_{i}, 1 ⩽ i ⩽ m, we denote by ⟨f_{1}, …, f_{m}⟩ : A → B_{1}×⋯ ×B_{m}

the map obtained by the Cartesian structure of Top. In particular, for the abovementioned i_{1}, …, i_{m} and for a simplicial space X: Δ^{op} → Top we have the map

p_{m} = ⟨i_{1}, …, i_{m}⟩: X_{m} → (X_{1})^{m}.

(According to our convention from the introduction, X(i_{k}) is abbreviated by i_{k}.) If m = 0, then (X_{1})^{0} = {∗} (a terminal object of Top) and let p_{0} denote the unique arrow from X_{0} to (X_{1})^{0}. The following lemma is claimed in [17].

Lemma 3.1. If X: Δ^{op} → Top is a simplicial space such that for every m ⩾ 0, the map p_{m} is a homotopy equivalence, then X_{1} is a homotopy associative H-space whose multiplication m is given by the composition

(X_{1})^{2} −−→^{p_{2}^{−1}} X_{2} −→^{d_{1}^{2}} X_{1},

where p_{2}^{−1} is an arbitrary homotopy inverse to p_{2}, and whose unit ∗ is s_{0}^{1}(x_{0}), for an arbitrary x_{0} ∈ X_{0}.

Definition 3.2. We say that a PRmss X : Δ^{op} → Top is Segal’s simplicial space when for every m ⩾ 0, the map p_{m} : X_{m} → (X_{1})^{m} is a homotopy equivalence.

Lemma 3.3. Let Y : Δ^{op} × Δ^{op} → Top be a PRmss. If for every k ⩾ 0, the simplicial space Y _{k} is Segal’s, then Y^{(1)} is Segal’s simplicial space.

Definition 3.4. We say that a PRmss X : (Δ^{op})^{n} → Top, where n ⩾ 1, is Segal’s multisimplicial space, when for every l ∈ {0, …, n − 1} and every k ⩾ 0, the simplicial space X_{^{1 … 1}_{l} k...k} is Segal’s.

Note that we do not require X_{k_{1} ...k_{l} k_{l+1} ...k_{n−1}} to be Segal’s for arbitrary k_{1}, …, k_{n−1} (see the parenthetical remark in Section 5.)

Remark 3.5. If X : (Δ^{op})^{n} → Top is Segal’s multisimplicial space, then for every l ∈ {0, …, n − 1}, X_{1...1} is homotopy associative H -space with respect to the structure obtained from Lemma 3.1 applied to X_{^{1 … 1}_{l} 1...1} : Δ^{op} → Top.

Our goal is to generalize the following proposition, which stems from [17, Proposition 1.5(b)]. (In the proof of that result, contractibility of |P A| comes from the fact that |P A| ≃ A_{0}.)

Proposition 3.6. Let X : Δ^{op} → Top be Segal’s simplicial space. If X_{1} with respect to the H-space structure obtained by Lemma 3.1 is grouplike, then X_{1} ≃ Ω|X|.

Our generalization is the following.

Proposition 3.7. Let X : (Δ^{op})^{n} → Top be Segal’s multisimplicial space. If X_{1...1}, with respect to the H-space structure obtained by Remark 3.5 when l = n−1, is grouplike, then X_{1...1} ≃ Ω^{n}|X|.

Proof. We proceed by induction on n ⩾ 1. If n = 1, the result follows from Proposition 3.6.

If n > 1, then we may apply the induction hypothesis to X_{1 ...} . Hence,

X_{1...1} ≃ Ω^{n−1} |X_{1 ...} |.

By Lemma 2.5, we have that |X_{1 ...} | ≈ | diagX_{1 ...} |. By the assumption and Remark 2.7(d), the multisimplicial space Y : Δ^{op} × Δ^{op} → Top, defined so that Y_{mk} = X_{mk...k} , satisfies the conditions of Lemma 3.3. Let Z be the simplicial space Y^{(1)} : Δ^{op} → Top, i.e.,

Z_{m} = |Y_{m} | = | diagX_{m ...} |.

By Lemma 3.3, Z is Segal’s simplicial space. By Remark 2.7(b), Z_{1} is a CWcomplex. Since the space Y_{10} (i.e., X_{10...0}) is by the assumption homotopic to (X_{110...0} )^{0}, it is contractible, and hence, path-connected. By Lemma 2.14, we have that Z_{1}, which is equal to |Y_{1} |, is path-connected. Note also that |Z| = |Y | ≈ | diagX| ≈ |X|.

By Lemma 3.1, Z_{1} is a homotopy associative H-space, and since it is a pathconnected CW-complex, by [1, Proposition 8.4.4], it is grouplike. Applying Proposition 3.6 to Z, we obtain

|X_{1 ...} | ≈ | diagX_{1 ...} | = Z_{1} ≃ Ω|Z| ≈ Ω|X|.

Finally, we have

X_{1...1} ≃ Ω^{n−1} |X_{1 ...} | ≃ Ω^{n}|X|.  ⊣

4. Segal’s lax functors

Thomason, [20], was the first who noticed that the reduced bar construction based on a symmetric monoidal category produces a lax, instead of an ordinary, functor. The idea to use Street’s rectification in that case, also belongs to him.

We use the notions of lax functor, left and right lax transformation as defined in [19]. The following theorem is taken over from [19, Theorem 2].

Theorem 4.1. For every lax functor W : A → Cat there exists a genuine functor V : A → Cat, a left lax transformation E : V → W and a right lax transformation J: W → V such that J is the left adjoint to E and W = EVJ.

We call V a rectification of W. It is easy to see that if W : A × B → Cat is a lax functor and V is its rectification, then for every object A of A, W_{A} is a lax functor and V_{A} is its rectification. As for simplicial spaces, for a (lax) functor W: Δ^{op} → Cat, we denote the unique arrow W_{0} → (W_{1})^{0} by p_{0}, and when m ⩾ 1, we have p_{m} = ⟨i_{1}, …, i_{m}⟩: W_{m} → (W_{1})^{m}.

Definition 4.2. We say that a lax functor W : Δ^{op} → Cat is Segal’s, when for every m ⩾ 0, p_{m} : W_{m} → (W_{1})^{m} is the identity. We say that a lax functor W : (Δ^{op})^{n} → Cat is Segal’s, when for every l ∈ {0, …, n − 1} and every k ⩾ 0, the lax functor W_{^{1 … 1}_{l} k...k} : Δ^{op} → Cat is Segal’s.

We denote by B: Cat → Top the classifying space functor, i.e., the composition | | ∘ N, where N: Cat → Top^{Δ^{op}} is the nerve functor.

Proposition 4.3. If W : Δ^{op} → Cat is Segal’s lax functor and V is its rectification, then B ∘ V is Segal’s simplicial space.

By Definitions 3.4 and 4.2, the following generalization of Proposition 4.3 is easily obtained.

Corollary 4.4. If W : (Δ^{op})^{n} → Cat is Segal’s lax functor and V is its rectification, then B ∘ V is Segal’s multisimplicial space.

For Corollary 4.4, we conclude that B ∘ V is a PRmss as in the proof of Proposition 4.3 given in the appendix.

5. An application

Let M be an n-fold strict monoidal category and let \overline{W}M: (Δ^{op})^{n} → Cat be

the n-fold reduced bar construction defined as in [5]. The main result of that paper says that \overline{W}M is a lax functor and it is easy to verify that it is Segal’s. (Note that \overline{W}M_{k_{1} ...k_{l} ...k_{n−1}} is not Segal’s when k_{j} > 1, for some 1 ⩽ j ⩽ l.)

For V being a rectification of \overline{W}M, we have the following.

Theorem 5.1. If BV_{1...1}, with respect to the H-space structure obtained by Remark 3.5 when l = n − 1, is grouplike, then BM ≃ Ω^{n}|B ∘ V |.

Proof. By Corollary 4.4, we have that B ∘ V is Segal’s multisimplicial space. Hence, by Proposition 3.7, BV_{1...1} ≃ Ω^{n}|B ∘ V |. Since V is a rectification of \overline{W}M, by relying on Remark A1 of the appendix, we conclude that BV_{1...1} ≃ B\overline{W}M_{1...1}. From the fact that \overline{W}M_{1...1} = M, we obtain that BM ≃ Ω^{n}|B ∘ V |. ⊣

This means that up to group completion (see [17] and [13]), for every n-fold strict monoidal category M, the classifying space BM is an n-fold loop space. When M contains a terminal or initial object, we have that BM, and hence BV_{1...1}, is path-connected. In that case, by [1, Proposition 8.4.4], BV_{1...1} is grouplike, and |B ∘ V | is an n-fold delooping of BM.

6. Appendix

Proof of Lemma 3.1. First, we prove that ⟨X_{1}, m, ∗⟩ is an H-space. Let j_{1} : X_{1} → X_{1} × X_{1} be such that j_{1}(x) = (x, ∗), and analogously, let j_{2} : X_{1} → X_{1} × X_{1} be such that j_{2}(x) = (∗, x). By the assumption, X_{0} is contractible. Hence, d_{0}^{1} is homotopic to the constant map to x_{0} and therefore s_{0}^{1} ∘ d_{0}^{1} is homotopic to the constant map to ∗. We conclude that

j_{1} ≃ ⟨1_{X_{1}} , s_{0}^{1} ∘ d_{0}^{1}⟩ = ⟨d_{2}^{2} ∘ s_{1}^{2}, d_{0}^{2} ∘ s_{1}^{2}⟩ = ⟨d_{2}^{2}, d_{0}^{2}⟩ ∘ s_{1}^{2} = p_{2} ∘ s_{1}^{2},

i.e., p_{2}^{−1} ∘ j_{1} ≃ s_{1}^{2}. Hence,

m ∘ j_{1} = d_{1}^{2} ∘ p_{2}^{−1} ∘ j_{1} ≃ d_{1}^{2} ∘ s_{1}^{2} = 1_{X_{1}} .

Analogously, m ∘ j_{2} ≃ 1_{X_{1}} and we have that ⟨X_{1}, m, ∗⟩ is an H-space.

Next, we prove that m is associative up to homotopy, i.e., that

m ∘ (m × 1) ≃ m ∘ (1 × m).

Consider p_{3} : X_{3} → (X_{1})^{3} for which we have

p_{3} = ⟨⟨i_{1}, i_{2}⟩, i_{3}⟩ = ⟨⟨d_{2}^{2} ∘ d_{3}^{3}, d_{0}^{2} ∘ d_{3}^{3}⟩, i_{3}⟩ = ⟨p_{2} ∘ d_{3}^{3}, i_{3}⟩
= (p_{2} × 1) ∘ ⟨d_{3}^{3}, i_{3}⟩, and analogously
p_{3} = (1 × p_{2}) ∘ ⟨i_{1}, d_{0}^{3}⟩.

Since p_{2} and p_{3} are homotopy equivalences, we have that ⟨d_{3}^{3}, i_{3}⟩ and ⟨i_{1}, d_{0}^{3}⟩ are homotopy equivalences, too. Moreover,

(1)  ⟨d_{3}^{3}, i_{3}⟩^{−1} ≃ p_{3}^{−1} ∘ (p_{2} × 1),
(2)  ⟨i_{1}, d_{0}^{3}⟩^{−1} ≃ p_{3}^{−1} ∘ (1 × p_{2}).

Also, we show that

(3)  d_{1}^{2} × 1 ≃ p_{2} ∘ d_{1}^{3} ∘ p_{3}^{−1} ∘ (p_{2} × 1),
(4)  1 × d_{1}^{2} ≃ p_{2} ∘ d_{2}^{3} ∘ p_{3}^{−1} ∘ (1 × p_{2}).

We have

(d_{1}^{2} × 1) ∘ ⟨d_{3}^{3}, i_{3}⟩ = ⟨d_{1}^{2} ∘ d_{3}^{3}, i_{3}⟩ = ⟨d_{2}^{2} ∘ d_{1}^{3}, d_{0}^{2} ∘ d_{1}^{3}⟩ = ⟨d_{2}^{2}, d_{0}^{2}⟩ ∘ d_{1}^{3} = p_{2} ∘ d_{1}^{3},

which together with (1) delivers (3). Also,

(1 × d_{1}^{2}) ∘ ⟨i_{1}, d_{0}^{3}⟩ = ⟨i_{1}, d_{1}^{2} ∘ d_{0}^{3}⟩ = ⟨d_{2}^{2} ∘ d_{2}^{3}, d_{0}^{2} ∘ d_{2}^{3}⟩ = ⟨d_{2}^{2}, d_{0}^{2}⟩ ∘ d_{2}^{3} = p_{2} ∘ d_{2}^{3},

which together with (2) delivers (4). Finally, we have

m ∘ (m × 1) = d_{1}^{2} ∘ p_{2}^{−1} ∘ (d_{1}^{2} × 1) ∘ (p_{2}^{−1} ) ≃ d_{1}^{2} ∘ d_{1}^{3} ∘ p_{3}^{−1} , by (3)
= d_{1}^{2} ∘ d_{2}^{3} ∘ p_{3}^{−1} ≃ d_{1}^{2} ∘ p_{2}^{−1} ∘ (1 × d_{1}^{2}) ∘ (1 × p_{2}^{−1} ), by (4)
= m ∘ (1 × m).  ⊣

Proof of Lemma 3.3. Let Z: Δ^{op} → Top be Y^{(1)}. By Remark 2.7(a), it is a PRmss. We have to show that for every m ⩾ 0, the map p_{m} : Z_{m} → (Z_{1})^{m} is a homotopy equivalence.

Let m = 0 and let T be the trivial simplicial space with T_{k} = {∗}. Consider the simplicial space Y_{0} : Δ^{op} → Top, which is a PRmss by Remark 2.7(c). By Proposition 2.9 and Lemma 2.10, both T and Y_{0} are proper. The following simplicial map is obtained by the assumptions (the diagrams are commutative since {∗} is terminal).

↓ ≃  ↓ ≃  ↓ ≃ … Y_{02}  → → → ← ←  Y_{01}  → → ←  Y_{00} … {∗}  → ←  {∗}  → ←  {∗} Y_{0} : T :

By Lemma 2.11, we have that |Y_{0} | ≃ |T | = {∗} via the unique map. Since Z_{0} = |Y_{0} | and (Z_{1})^{0} = {∗}, we are done.

Let m > 0. Consider the simplicial spaces Y_{m} and (Y_{1} )^{m}, which are proper by Remark 2.7(c), Proposition 2.9, Lemma 2.10 and Remark 2.13. The following simplicial map is obtained by the assumptions (it is straightforward to verify that the diagrams are commutative).

↓ ≃  ↓ ≃  ↓ ≃ … Y_{m2}  → → → ← ←  Y_{m1}  → → ←  Y_{m0} … (Y_{12})^{m}  → → → ← ←  (Y_{11})^{m}  → → ←  (Y_{10})^{m} Y_{m} : (Y_{1} )^{m} :

By Lemma 2.11, we have that

|⟨Y (i_{1}, ), …, Y (i_{m}, )⟩|: |Y_{m} | → |(Y_{1} )^{m}|

is a homotopy equivalence. Also, for Top, the realization functor | | preserves products (see [10, Theorem 14.3], [7, III.3, Theorem] and [11, Corollary 11.6]). Namely, for π_{k}: (Y_{1} )^{m} → Y_{1} , 1 ⩽ k ⩽ m being the kth projection,

⟨|π_{1}|, …, |π_{m}|⟩: |(Y_{1} )^{m}| → |Y_{1} |^{m}

is a homeomorphism (| | is strong monoidal; see [16, Example 6.2.2]). Hence,

⟨|π_{1}|, …, |π_{m}|⟩ ∘ |⟨Y (i_{1}, ), …, Y (i_{m}, )⟩|: |Y_{m} | → |Y_{1} |^{m}

is a homotopy equivalence.

The following easy computation, in which ⟨Y (i_{1}, ), …, Y (i_{m}, )⟩ is abbreviated by α,

⟨|π_{1}|, …, |π_{m}|⟩ ∘ |α| = ⟨|π_{1}| ∘ |α|, …, |π_{m}| ∘ |α|⟩ = ⟨|π_{1} ∘ α|, …, |π_{m} ∘ α|⟩
= ⟨|Y (i_{1}, )|, …, |Y (i_{m}, )|⟩ = ⟨Z(i_{1}), …, Z(i_{m})⟩

shows that the map p_{m} = ⟨Z(i_{1}), …, Z(i_{m})⟩ is a homotopy equivalence between Z_{m} = |Y_{m} |, and (Z_{1})^{m} = |Y_{1} |^{m}. ⊣

Some preliminary remarks for Proposition 4.3. Let 2 be the category with two objects (0 and 1) and one nonidentity arrow h: 0 → 1. Let I_{0}, I_{1} : C → C×2 be the functors such that for every object C of C, we have that I_{0}(C) = (C, 0) and I_{1}(C) = (C, 1). Let F, G: C → D be two functors. There is a bijection between the set of natural transformations α: F →^{.} G, and the set of functors A: C×2 → D such that A ∘ I_{0} = F and A ∘ I_{1} = G. This bijection maps α: F →^{.} G to A: C × 2 → D such that

A(C, 0) = F C, A(C, 1) = GC, A(f, 1_{0}) = F f, A(f, 1_{1}) = Gf,

and for f: C → C′,

A(f, h) = Gf ∘ α_{C} = α_{C′} ∘ F f.

Its inverse maps A: C × 2 → D to α: F →^{.} G such that α_{C} = A(1_{C}, h).

The nerve functor N preserves products on the nose, hence, the classifying space functor B = | | ∘ N preserves products too. Therefore, the spaces BC × I (i.e., BC × B2) and B(C × 2) are homeomorphic and we have the following.

Remark A1. Every natural transformation α: F →^{.} G gives rise to the homotopy

BC × I ⟶^{≈} B(C × 2) −−→^{BA} BD

between the maps BF and BG.

Proof of Proposition 4.3. By the isomorphism mentioned in Definition 2.2, we have that N ∘ V corresponds to a multisimplicial set X: Δ^{op} × Δ^{op} → Top and B ∘ V is X^{(1)}. Hence, it is a PRmss.

We have to show that for every m ⩾ 0, p_{m} : BV_{m} → (BV_{1})^{m} is a homotopy equivalence, where we denote again by p_{0} the unique map from BV_{0} to (BV_{1})^{0} and by p_{m} the map ⟨BV (i_{1}), …, BV (i_{m})⟩.

When m = 0, we show that BJ_{0} : BW_{0} → BV_{0} is a homotopy inverse to p_{0}. Since W_{0} and (V_{1})^{0} are the same trivial category and BW_{0} = (BV_{1})^{0} = {∗}, it is easy to conclude that p_{0} ∘ BJ_{0} ≃ 1_{(BV_{1} )^{0}} , and that p_{0} = BE_{0}. The latter, by the adjunction J_{0} ⊣ E_{0} and Remark A1, delivers BJ_{0} ∘ p_{0} ≃ 1_{BV_{0}} .

When m ⩾ 1, we have for every 1 ⩽ j ⩽ m, the following natural transformations.

[figure: V_{m}; V_{1}; W_{1}; W_{m}; i_{j}; E_{m}; i_{j}; E_{1}; ⇓; E_{i_{j}}; W_{m}; W_{1}; V_{1}; V_{m}; i_{j}; J_{m}; i_{j}; J_{1}; ⇑; J_{i_{j}}]

By using the monoidal structure of Cat given by 2-products and the fact that ⟨i_{1}, …, i_{m}⟩ : W_{m} → (W_{1})^{m} is the identity, we obtain the following two natural transformations.

[figure: V_{m}; (V_{1})^{m}; W_{m}; ⟨i_{1}, …, i_{m}⟩; E_{m}; (E_{1})^{m}; ⇓; ⟨E_{i_{1}} , …, E_{im} ⟩; W_{m}; (V_{1})^{m}; V_{m}; ⟨i_{1}, …, i_{m}⟩; (J_{1})^{m}; J_{m}; ⇑; ⟨J_{i_{1}} , …, J_{im} ⟩]
For π_{k}: C^{m} → C, 1 ⩽ k ⩽ m being the kth projection,
⟨Bπ_{1}, …, Bπ_{m}⟩: BC^{m} → (BC)^{m}

is a homeomorphism whose inverse we denote by q_{m}(C). It is easy to verify that for F, F_{1}, …, F_{m} : C → D we have B⟨F_{1}, …, F_{m}⟩ = q_{m}(D)⟨BF_{1}, …, BF_{m}⟩ and BF^{m} ∘ q_{m}(C) = q_{m}(D) ∘ (BF)^{m}.

By Remark A1, the transformations mentioned above give rise to

(†)  BE_{m} ≃ B(E_{1})^{m} ∘ B⟨V (i_{1}), …, V (i_{m})⟩
= q_{m}(W_{1}) ∘ (BE_{1})^{m} ∘ ⟨BV (i_{1}), …, BV (i_{m})⟩
= q_{m}(W_{1}) ∘ (BE_{1})^{m} ∘ p_{m},
(††)  B(J_{1})^{m} ≃ B⟨V (i_{1}), …, V (i_{m})⟩ ∘ BJ_{m}
= q_{m}(V_{1}) ∘ ⟨BV (i_{1}), …, BV (i_{m})⟩ ∘ BJ_{m}
= q_{m}(V_{1}) ∘ p_{m} ∘ BJ_{m}.

The following calculation shows that

BJ_{m} ∘ q_{m}(W_{1}) ∘ (BE_{1})^{m} : (BV_{1})^{m} → BV_{m}

is a homotopy inverse to p_{m}.

1_{BV_{m}} ≃ BJ_{m} ∘ BE_{m}, by J_{m} ⊣ E_{m}, Remark A1
≃ BJ_{m} ∘ q_{m}(W_{1}) ∘ (BE_{1})^{m} ∘ p_{m}, by (†)
1_{(BV_{1} )^{m}} ≃ q_{m}^{−1} (V_{1}) ∘ B(J_{1})^{m} ∘ B(E_{1})^{m} ∘ q_{m}(V_{1}), by J_{1} ⊣ E_{1}, Remark A1
≃ p_{m} ∘ BJ_{m} ∘ q_{m}(W_{1}) ∘ (BE_{1})^{m}, by (††).  ⊣

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Mathematical Institute (Received 18 07 2014) SANU (Revised 23 09 2014) Belgrade Serbia

 zpetric@mi.sanu.ac.rs