Strongly Homotopy-Commutative

Monoids Revisited

Michael Brinkmeier

Received: September 22, 1999 Revised: November 11, 2000

Communicated by Günter M. Ziegler

Abstract. We prove that the delooping, i. e., the classifying space, of a grouplike monoid is an H-space if and only if its multiplication is a homotopy homomorphism, extending and clarifying a result of Sugawara. Furthermore it is shown that the Moore loop space functor and the construction of the classifying space induce an adjunction of the according homotopy categories.

2000 Mathematics Subject Classification: Primary 55P45, 55P35; Secondary 55R35 Keywords and Phrases: H-spaces, Classifying space, Monoid, Strongly homotopy commutative, Homotopy homomorphism

Introduction

In [Sug60] Sugawara examined structures on topological monoids, which induce H-space multiplications on the classifying spaces. He introduced a form of coherently homotopy commutative monoids, which he called strongly homotopy commutative. His main result is that a countable CW -group G is strongly homotopy-commutative if and only if its classifying space BG is an H-space. The proof proceeds as follows. One first shows that the multiplication G × G → G of a strongly homotopy commutative group is a homotopy homomorphism (Sugawara called such maps strongly homotopy multiplicative), i.e. a homomorphism up to coherent homotopies. Then one shows that this map induces an H-space structure on BG. The proof of the converse is very sketchy and far from convincing. We start with an easy to handle reformulation of the notion of homotopy homomorphisms. The well-pointed and grouplike monoids (cmp. Def. 2.4) and homotopy classes of these homotopy homomorphisms form a category HGr_{H}. If 𝔗𝔬𝔭_{H}^{∗} is the category of well-pointed spaces and based homotopy classes of maps, then the classifying space and the Moore loop space functors induces functors B_{H} : HGr_{H} → 𝔗𝔬𝔭_{H}^{∗} and Ω_{H} : 𝔗𝔬𝔭_{H}^{∗} → HGr_{H}. We first prove the following strengthening of a result of Fuchs ([Fuc65]).

Theorem (3.7). The functor B_{H} is left adjoint to Ω_{H}. The adjunction induces an equivalence of the full subcategories of monoids in HGr_{H} of the homotopy type of CW -complexes and of the full subcategory of 𝔗𝔬𝔭_{H}^{∗} of connected spaces of the homotopy type of CW -complexes.

We then reexamine Sugawara’s result starting with grouplike monoids whose multiplications are homotopy homomorphisms. They give rise to H-objects (i.e. Hopf objects) in the category HGr_{H}. We obtain the following extension of Sugawara’s theorem.

Theorem (3.8 and 4.2). The classifying space of a grouplike and well-pointed monoid M is an H-space if and only if M is an H-object in HGr_{H}.

As mentioned above the multiplication of a strongly homotopy commutative monoid is a homotopy homomorphism. We were not able to prove the converse and consider it an open question. I would like to thank Rainer Vogt for his guidance and help during the preparation of this paper, and James Stasheff for his corrections and suggestions. The author was supported by the Deutsche Forschungsgemeinschaft.

1 The W-construction

Let Mon be the category of well-pointed, topological monoids and continuous homomorphisms between them. Here well-pointed means, that the inclusion of the unit is a closed cofibration.

Remark 1.1. One can functorially replace any monoid M by well-pointed one by adding a whisker (cmp. [BV68], pg 1130f.). This does not change the (unbased) homotopy type of M.

Definition 1.2. Let M and N be topological monoids. A homotopy H_{t} : M → N is called a homotopy through homomorphisms if for each t ∈ I the map H_{t} : M → N is a homomorphism.

Definition 1.3. (cmp. [BV73], [Vog73], [SV86]) We define a functor W : Mon → Mon. For M ∈ 𝔬𝔟 Mon the monoid W M is the space

W M = ∐_{n∈ℕ} M^{n+1} × I^{n}/ ∼

Strongly Homotopy-Commutative Monoids Revisited 615

with the relation

(x_{0}, t_{1}, x_{1}, …, t_{n}, x_{n}) =

(x_{0}, …, t_{i−1}, x_{i−1}x_{i}, t_{i+1}, …, x_{n})  for t_{i} = 0
(x_{1}, t_{2}, …, x_{n})  for x_{0} = e
(x_{0}, …, x_{i−1}, max(t_{i}, t_{i+1}), x_{i+1}, …, x_{n}) for x_{i} = e
(x_{0}, …, t_{n−1}, x_{n−1})  for x_{n} = e.

The multiplication is given by

(x_{0}, …, t_{n}, x_{n}) · (y_{0}, s_{1}, …, y_{k}) = (x_{0}, …, t_{n}, x_{n}, 1, y_{0}, s_{1}, …, y_{k}).

A continuous homomorphism F : M → N is mapped to W F : W M → W N with

W F (x_{0}, t_{1}, x_{1} …, x_{n}) = (F (x_{0}), t_{1}, F (x_{1}), …, F (x_{n}))).

The augmentation ε_{M} : W M → M with ε_{M}(x_{0}, …, x_{n}) = x_{0} · · · · · x_{n} defines a natural transformation ε : W → id. If i_{M} : M → W M is the inclusion, which maps every element x of M to the chain (x), we get ε_{M} ∘ i_{M} = id_{M} and a non-homomorphic homotopy h_{t} : W M → W M from i_{M} ∘ ε_{M} to id_{M}, given by

h_{t}(x_{0}, t_{1}, x_{1}, …, t_{n}, x_{n}) = (x_{0}, tt_{1}, x_{1}, …, tt_{n}, x_{n}).

Therefore ε_{M} is a homotopy equivalence and M a strong deformation retract of W M at space level, i.e. its homotopy inverse is no homomorphism. One of the most important properties of the W -construction is the following lifting theorem, which is a slight variation of [SV86, 4.2] and is proven in the same way.

Theorem 1.4. Given the following diagram in Mon with 0 ≤ n ≤ ∞ such that

\xymatrix{W M \ar[dr]_{F} \ar@{.>}[rr]^{H} & & B \ar[dl]^{L} \\ & N}

1. M is well-pointed and

2. L is a homotopy equivalence.

Then there exists a homomorphism H : W M → B and a homotopy K_{t} : W M → N through homomorphisms from L ∘ H to F. Furthermore H is unique up to homotopy through homomorphisms.

2 Homotopy homomorphisms

Definition 2.1. Let M and N be two well-pointed monoids. A homotopy homomorphism F from M to N is a homomorphism F : W M → W N. The map f := ε_{N} ∘ F ∘ i_{M} : M → N is the underlying map of F. Let HMon be the category whose objects are well-pointed, topological monoids, and whose morphisms are homotopy homomorphisms.

Remark 2.2. Our homotopy homomorphisms are closely related to Sugawara’s approach. If we compose a homotopy homomorphism with the augmentation, we obtain a map W M → N which is, up to the conditions for the unit, a strong homotopy multiplicative map in Sugawara’s sense. Since ε_{N} is a homotopy equivalence, the resulting structures are equivalent, after passage to the homotopy category.

The Moore loop-space construction Ω_{M}X and the classifying space functor B define functors Ω_{W} : 𝔗𝔬𝔭^{∗} → HMon and B_{W} : HMon → 𝔗𝔬𝔭^{∗} by Ω_{W} (X) = Ω_{M}X and B_{W} (M) = B(W M) on objects and Ω_{W} (f) = W Ω_{M}f and B_{W} (F) = BF on morphisms. For a based map f : X → Y let [f]_{∗} denote its based homotopy class. For a homomorphism F of monoids let [F] denote its homotopy class with respect to homotopies through homomorphisms. Let 𝔗𝔬𝔭_{H}^{∗} be the category of based, well-pointed spaces and based homotopy classes of based spaces and HMon_{H} the category of well-pointed monoids and homotopy classes of homotopy homomorphisms.

Remark 2.3. One can prove that the homotopy homomorphisms, which are homotopy equivalences on space level, represent isomorphisms in HMon_{H}.

Since Ω_{W} and B_{W} preserve homotopies, they induce a pair of functors.

B_{H} : 𝔗𝔬𝔭_{H}^{∗} ⇄ HMon_{H} : Ω_{H}

Definition 2.4. A monoid M with multiplication μ and unit e is called grouplike, if there a continuous map i : M → M such that the maps x ↦ μ(x, i(x)) and x ↦ μ(i(x), x) are homotopic to the constant map on e.

Since the Moore loop-spaces are grouplike and since this notion is homotopy invariant, an additional restriction is necessary for Theorem 3.7 to be true. Let HGr be the full subcategory of HMon, whose objects are grouplike, and let HGr_{H} be the corresponding homotopy category. Then B_{H} and Ω_{H} give rise to a pair of functors

B_{H} : 𝔗𝔬𝔭_{H}^{∗} ⇄ HGr_{H} : Ω_{H}.

We make use of a construction from [SV86]. For an arbitrary monoid M let EM be the contractible space with right M-action such that EM/M ≃ BM. We define a monoid structure on the Moore path space

P (EM; e, M) :=

{(ω, l) ∈ EM^{ℝ_{+}} × ℝ_{+} : ω(0) = e, ω(l) ∈ M, ω(t) = ω(l) for t ≥ l} .

The product of two paths (ω, l) and (ν, k) is given by (ρ, l + k), with

ρ(t) = \begin{cases}ω(t) & if 0 ≤ t ≤ l \\ ω(l) · ν(t − l) & if l ≤ t ≤ l + k. \end{cases}

The end-point projection π_{M} : P (EM; e, M) → M, (ω, l) ↦ ω(l) a continuous homomorphism. Since P (EM; e, M) is the homotopy fiber of the inclusion i : M ↪ EM and since EM is contractible, π_{M} is a homotopy equivalence. By Theorem 1.4 there exists a homomorphism T̄_{M} : W M → P (EW M; e, W M) such that the following diagram commutes up to homotopy through homomorphisms.

[figure: W M; T̄_{M}; P (EW M; e, W M); π_{W M}; W M]

Because π_{W M} is strictly natural in W M, T̄_{M} is natural up to homotopy through homomorphism. Obviously we have P (BW M, ∗, ∗) = Ω_{M}BW M. Hence the projection p_{W M} : EW M → BW M induces a natural homomorphism P (p_{W M}) : P (EW M; e, W M) → Ω_{M}BW M. Because W M is grouplike, P (p_{W M}) is a homotopy equivalence. Therefore we obtain a homomorphism T_{M} : W M → W Ω_{M}BW M, which is induced by Theorem 1.4 and the following diagram.

\xymatrix{W M \ar[r]^{T_{M}} \ar[d]_{T̄_{M}} & W Ω_{M}BW M \ar[d]^{ε_{Ω_{M} BW M}} \\ P (EW M; e, W M) \ar[r]_{P (p_{M})} & Ω_{M}BW M}

Since all morphisms are natural up to homotopy through homomorphisms, the T_{M} form a natural transformation [T] from id_{HGr_{H}} to Ω_{H}B_{H} and each T_{M} is a homotopy equivalence and hence an isomorphism in HGr_{H}. Its inverse [K_{M}] can be constructed by Theorem 1.4 and the following diagram.

\xymatrix{W Ω_{M}BW M \ar@{-->}[rr]^{K_{M}} &  & W M \ar[dl]^{T_{M}} \\  & W Ω_{M}BW M}

For each well-pointed space X, we chose E_{X} to be the dotted arrow in the following diagram.

\xymatrix{BW Ω_{M}BW Ω_{M}X \ar[r]^{BK_{Ω_{M} X}} \ar[d]_{Bε_{Ω_{M} BW Ω_{M} X}} & BW Ω_{M}X \ar[d]^{Bε_{Ω_{M} X}} \\ BΩ_{M}BW Ω_{M}X \ar[d]_{e_{BW Ω_{M} X}} & BΩ_{M}X \ar[d]^{e_{X}} \\ BW Ω_{M}X \ar@{.>}[r]_{E_{X}} & X}

Here the e_{•} are the maps described in Proposition 5.1. Since all solid arrows, except for e_{X}, are based homotopy equivalences the morphism E_{X} exists and is uniquely determined up to based homotopy. The naturality of E_{X} follows from the naturality up to homotopy of all other maps. Hence we have a natural transformation [E]_{∗} from B_{H}Ω_{H} to the identity on 𝔗𝔬𝔭_{H}^{∗}.

Theorem 2.5. The functor B_{H} : HGr_{H} → 𝔗𝔬𝔭_{H}^{∗} is left adjoint to Ω_{H}. The natural isomorphism [T] is the unit, and the natural transformation [E]_{∗} the counit of this adjunction.

Proof. The definition of E_{BW M} and the naturality of several morphisms imply

[E_{BW M} ∘ BT_{M} ∘ e_{BW M}]_{∗} = [e_{BW M}]_{∗}

and since e_{BW M} is a based homotopy equivalence by Proposition 5.1 this results in

[E_{B_{H}(M)}]_{∗} ∘ B_{H}[T_{M}] = [E_{BW M}]_{∗} ∘ [BT_{M}]_{∗} = [id_{BM}]_{∗}.

The definition of E_{X} implies

[W Ω_{M}E_{X} ∘ W Ω_{M}e_{BW Ω_{M} X} ∘ W Ω_{M}Bε_{Ω_{M} BW Ω_{M} X} ∘ W Ω_{M}BT_{Ω_{M} X}] = [W Ω_{M}e_{X} ∘ W Ω_{M}Bε_{Ω_{M} X}]

and the naturality of several maps leads to

[W Ω_{M}E_{X} ∘ W Ω_{M}e_{BW Ω_{M} X} ∘ W Ω_{M}Bε_{Ω_{M} BW Ω_{M} X} ∘ W Ω_{M}BT_{Ω_{M} X}] = [W Ω_{M}e_{X} ∘ W Ω_{M}Bε_{Ω_{M} X} ∘ W Ω_{M}BW Ω_{M}E_{X} ∘ W Ω_{M}BT_{Ω_{M} X}] .

Since ε_{Ω_{M} X} and Ω_{M}e_{X} are homotopy equivalences the homomorphisms W Ω_{M}e_{X} and W Ω_{B}ε_{Ω_{M} X} represent isomorphisms in HGr_{H}. Therefore we have

[W Ω_{M}BW Ω_{M}E_{X} ∘ W Ω_{M}BT_{Ω_{M} X}] = [id_{W Ω_{M} BW Ω_{M} X}] .

The facts that T_{Ω_{M} X} is an isomorphism in HGr_{H} and that

[T_{Ω_{M} X} ∘ W Ω_{M}E_{X} ∘ T_{Ω_{M} X}] =

[W Ω_{M}BW Ω_{M}E_{X} ∘ W Ω_{M}BT_{Ω_{M} X} ∘ T_{Ω_{M} X}]

imply

Ω_{H}[E_{X}]_{∗} ∘ [T_{Ω_{H}(X)}] = [W Ω_{M}E_{X} ∘ T_{Ω_{M} X}] = [id_{W Ω_{M} X}] .

3 Hopf-objects

Definition 3.1. An H- or Hopf-object (X, μ, ρ) in a monoidal category^{1} (C, ⊗, e) is a non-associative monoid, i.e. an object X of C together with morphisms μ : X ⊗ X → X and ρ : e → X such that the following diagram commutes.

\xymatrix{e ⊗ X \ar[r]^{ρ⊗id_{X}} \ar[dr]_{≃} & X ⊗ X \ar[d]^{μ} & X ⊗ e \ar[l]_{id_{X}⊗ρ} \ar[dl]^{≃} \\  & X.}

A morphism of H-objects (or H-morphism) f : X → Y is a morphism such that μ_{Y} ∘ (f ⊗ f) = f ∘ μ_{X}. The H-objects of C and the H-morphisms form a category HopfC.

Proposition 3.2. Let (C, ⊙, e_{C}) and (D, ⊗, e_{D}) be monoidal categories and

(F, G, η, ε) : C → D

an adjunction of monoidal functors^{2} such that the diagrams

\xymatrix{Y ⊙ Y \ar[r]_{η_{Y} ⊙η_{Y}} \ar[d]^{η_{Y ⊙Y}} & GF Y ⊙ GF Y \ar[d] & F GX ⊗ F GX \ar[r] \ar[d]_{ε_{X}⊗ε_{X}} & F (GX ⊙ GX) \ar[d] \\ GF (Y ⊙ Y) & G(F Y ⊗ F Y) \ar[l] & X ⊗ X & F G(X ⊗ X) \ar[l]}

commute for each X ∈ C and Y ∈ D, then there exists an adjoint pair of functors

HopfF : HopfHC ⇆ HopfD : HopfG.

Proof. HopfF is given by

HopfF (X, μ, ρ) = (F X, F μ ∘ φ, F ρ) and HopfF (f) = F f,

with φ : F X⊗F X → F (X⊙X) the natural transformation. Its adjoint HopfG is given analogously. The two commutative diagrams imply that the units η_{X} and the counits ε_{Y} of the adjunction are H-morphisms. Therefore they form the unit and counit of an adjunction.

Example 3.3. 𝔗𝔬𝔭_{H}^{∗} with its product is a monoidal category. The H-objects in 𝔗𝔬𝔭_{H}^{∗} are precisely the H-spaces with the base point as unit. The homotopy class [μ]_{∗} of the multiplication is called H-space structure of X. H-morphisms are the homotopy classes of H-space morphisms up to homotopy.

1For a definition of monoidal categories see [McL71].

2For a definition of monoidal functors see [BFSV98]

Documenta Mathematica 5 (2000) 613–624

Example 3.4. HGr_{H} has a monoidal structure ⊗ given on objects by M ⊗N = M × N. For morphisms F : W M → W M′ and G : W N → W N′ we define F ⊗ G : W (M × N) → W (M′ × N′) as follows: Let S_{M,N} = (W pr_{M}, W pr_{N}) : W (M × N) → W M × W N be induced by the two projections. Then the diagram

\xymatrix{W (M × N) \ar[rr]^{S_{M,N}} \ar[dr]_{ε_{M×N}} &  & W M × W N \ar[dl]^{ε_{M} ×ε_{N}} \\  & M × N.}

commutes. Obviously S_{M,N} is a homotopy equivalence. By Theorem 1.4 the homotopy class of S_{M,N} in HMon is uniquely determined. For two homotopy homomorphisms F : W M → W M′ and G : W N → W N′, we define F ⊗ G : W (M × N) → W (M′ × N′) to be the lifting in the following diagram.

\xymatrix{W (M × N) \ar[r]^{F ⊗G} \ar[d]_{S_{M,N}} & W (M′ × N′) \ar[d]^{S_{M′,N′}} \\ W M × W N \ar[r]_{F ×G} & W M′ × W N′.}

This construction is compatible with the composition and we can define a functor ⊗ : HGr_{H} × HGr_{H} → HGr_{H} with M ⊗ N = M × N and [F] ⊗ [G] = [F ⊗ G]. The projections [P_{M}] and [P_{N}] on M ⊗ N are given by [p_{i} ∘ S_{M,N}], where p_{i} is the according projection from W M × W N. It is easy to check that ⊗ and these projections form a product in HGr_{H} and that the trivial monoid ∗ is a terminal and initial object of HGr_{H}. Therefore HGr_{H} is monoidal and we have a notion of H-objects in HGr_{H}. The unit of an H-object in HGr_{H} is always the unit of the underlying monoid.

Lemma 3.5. If (M, [F]) is a H-object in HGr_{H}, then the underlying map f of F is homotopic to the multiplication μ of M.

Proof. The homomorphism F̄ = ε_{M} ∘ F has the property [F̄ ∘ W i_{k}] = [ε_{M}] for k = 1, 2. The homotopy h_{t} : M × M → M with h_{t}(x, y) = F̄((x, e), t, (e, y)) runs from f(x, y) to f(x, e)f(e, y), and hence f and μ are based homotopic.

Thus the multiplication μ of an H-object (M, [F]) in HGr_{H} is homotopic to the underlying map of F, and therefore homotopy-commutative with the commuting homotopy from xy to yx derived from F ((e, y), t, (x, e)). The relations in W (M × M) define higher homotopies so that the underlying monoid is homotopy commutative in a strong sense. We now want to examine the structure on a monoid M, that leads to the existence of an H-space multiplication on its classifying space.

Proposition 3.6. B_{H} and Ω_{H} are monoidal functors.

Proof. For M, N ∈ HGr_{H} the morphism

s_{M,N} : BW (M × N) → BW M × BW N

is given by the based homotopy equivalence (BW p_{1}, BW p_{2}), where p_{1}, p_{2} : M × M → M are the projections. For X, Y ∈ 𝔗𝔬𝔭_{H}^{∗} the morphism Ω_{H}(X × Y) ≃ Ω_{H}X ⊗ Ω_{H}Y is given by W (Ω_{M}p_{1}, Ω_{M}p_{2}) : W Ω_{M}(X × Y) → W (Ω_{M}X × Ω_{M}Y).

Theorem 3.2 now implies

Theorem 3.7. B_{H} and Ω_{H} induce an adjunction

HopfB_{H} : HopfHGr_{H} ⇆ Hopf𝔗𝔬𝔭_{H}^{∗} : HopfΩ_{H}

with

HopfB_{H}(M, [F]) = (BW M, [BF ∘ s_{M,M}]_{∗})

and

HopfΩ_{H}(X, [μ]_{∗}) = (Ω_{M}X, [W Ω_{M}μ ∘ R_{X,X}]).

Theorem 3.8. The classifying space BM of a grouplike and well-pointed monoid M is an H-space if and only if M is an H-object in HGr_{H}.

Proof. If M is an H-object, then BW M and thus BM are H-spaces. Now let BM be an H-space. Then Ω_{M}BW M is an H-object in HopfHGr_{H}. Since T_{M} : W M → W Ω_{M}BW M is a homotopy equivalence, M is an H-object, too.

4 Extensions

A monoid in Hopf𝔗𝔬𝔭_{H}^{∗} is a homotopy-associative H-space (X, μ). A monoid HopfHGr_{H} consists of a well-pointed and grouplike monoid together with homotopy homomorphisms F_{2} : W (M × M) → W M and F_{3} : W (M × M × M) → W M such that (M, [F_{2}]) is an H-object and

[F_{2} ∘ (F_{2} ⊗ id)] = [F_{3}] = [F_{2} ∘ (id ⊗ F_{2})].

We call the H-object (M, [F_{2}]) associative. Since these structures are invariant under isomorphisms we obtain, similar to the non-associative case, the following

Theorem 4.1. The classifying space BM of a well-pointed, grouplike monoid M is an homotopy associative H-space, if M is an associative H-object in HGr_{H}.

As we realized earlier, the morphism e_{X} : BΩ_{M}X → X need not be a homotopy equivalence. But by Proposition 5.1 Ω_{M}e_{X} is a based homotopy equivalence. Hence, if we restrict to connected, based spaces of the homotopy type of CW - complexes, e_{X} is a homotopy equivalence. This implies that the adjunction

B_{H} : HGr_{H} ⇆ 𝔗𝔬𝔭_{H}^{∗} : Ω_{H}

induces an equivalence of categories, if we restrict to the full subcategories of based spaces of the homotopy type of connected CW-complexes and grouplike monoids of the homotopy type of CW -complexes.

Theorem 4.2. The full subcategories HopfHGr_{H}^{CW} ⊂ HopfHGr_{H} of Hobjects of the homotopy type of CW-complexes, and Hopf𝔗𝔬𝔭_{H}^{∗,CW} ⊂ Hopf𝔗𝔬𝔭_{H}^{∗} of connected H-spaces of the homotopy type of CW -complexes, are equivalent.

5 Appendix: The evaluation map

This section is dedicated to the proof of the following theorem.

Proposition 5.1. For each based space X there exists a natural map e_{X} : BΩ_{M}X → X such that

1. Ω_{M}e_{X} is a homotopy equivalence for each based space X and

2. if M is a grouplike well-pointed monoid then e_{BM} is a homotopy equivalence.

To prove this we will use based simplicial spaces. A based simplicial space is a functor from the dual of the category Δ of finite, ordered sets [n] = {0, 1, …, n} to 𝔗𝔬𝔭_{∗}. The based standard simplices ∇_{∗}(n) are given by the quotient space ∇(n)/V_{n} with ∇(n) the n-th standard simplex and V_{n} its subspace of vertices. They induce a based cosimplicial space ∇_{∗} : Δ → 𝔗𝔬𝔭_{∗}. We define the based geometric realization of a based simplicial space 𝔛 as

| · |_{∗} = ∐_{n} 𝔛(n) ∧ ∇_{∗}(n)/ ∼

with the relation ∼ generated by the same equalities as in the unbased case. This induces a functor |·|_{∗} from the category of based simplicial spaces to 𝔗𝔬𝔭_{∗}. Analogous to the unbased singular complex we can define the based singular complex S_{∗}X : Δ^{op} → 𝔗𝔬𝔭_{∗} of a based space X by

[n] ↦ 𝔗𝔬𝔭_{∗}(∇_{∗}(n), X).

S_{∗} induces a functor from 𝔗𝔬𝔭_{∗} to the category of based simplicial sets. As in the unbased case this right adjoint to the based realization | · |_{∗}. The unit τ_{∗} : id → S_{∗}| · |_{∗} is given by

τ_{∗,𝔛}(x) = (t ↦ (x, t)) , x ∈ 𝔛_{n}, t ∈ ∇_{∗}(n) and the counit η_{∗} : |S_{∗} · |_{∗} → id by

η_{∗,X}(ω, t) = ω(t), ω ∈ S_{∗}Y (n), t ∈ ∇_{∗}(n).

Definition 5.2. (cmp. [Seg74, A.4.]) A based simplicial space 𝔛 is good if for each n and 0 ≤ i ≤ n the inclusion s_{i}(𝔛_{n−1}) ↪ 𝔛_{n} is a closed cofibration.

Now observe that the based realization |𝔛|_{∗} coincides with the unbased realization |𝔛| if the simplicial space 𝔛 has only one 0-simplex. Therefore we obtain the following lemma from well-known facts.

Lemma 5.3. (cmp. [Seg74, A.1]) Let 𝔛 and 𝔜 be good, based simplicial spaces with 𝔛_{0} = ∗ = 𝔜_{0} and let 𝔣 : 𝔛 → 𝔜 be a based simplicial map. If each map 𝔣_{n} is a based homotopy equivalence, then the map

|𝔣|_{∗} : |𝔛|_{∗} → |𝔜|_{∗}

is a based homotopy equivalence.

In the following we will show that the nerve Ω_{M}^{•}X of the Moore loop space of an arbitrary well-pointed space X is homotopy equivalent to its based simplicial complex. There exists a based simplicial map a : Ω_{M}^{•}X → S_{∗}X, given by

a_{n}(ω_{1}, …, ω_{n})(t_{0}, …, t_{n}) = (ω_{1} + ⋯ + ω_{n}) (∑_{i=1}^{n} ∑_{j=1}^{i} t_{i}l_{j})

(l_{j} is the length of the loop ω_{j} and + the loop addition). Let 𝔢_{j} = (t_{0}, …, t_{n}) be the vertex of ∇(n) given by t_{j} = 1, t_{k} = 0, k ≠ j. Then a maps the loop ω_{j} to the edge running from 𝔢_{j−1} to 𝔢_{j}. E_{n} := {(t_{0}, …, t_{n}) ∈ ∇(n) : t_{i} + t_{i+1} = 1 for some i} is a strong deformation retract of ∇(n) and there exists a sequence of homotopy equivalences

𝔗𝔬𝔭_{∗} (∇_{∗}(n), X) ≃ 𝔗𝔬𝔭_{∗} (E_{n}, X) ≃ (ΩX)^{n} ≃ (Ω_{M}X)^{n}

such that the composition of a with these maps is the endomorphism of (Ω_{M}X)^{n} which changes the length of the loops to length 1. This map is homotopic to the identity, and hence a is a homotopy equivalence. Furthermore a is natural in X and defines a natural transformation from Ω_{M}^{•} to S_{∗}. If X and hence Ω_{M}X and 𝔗𝔬𝔭_{∗} (∇_{∗}(n), X) are well-pointed, then a_{X} is a based homotopy equivalence. The map e_{X} := η_{∗,X} ∘ |a_{X}|_{∗} : |Ω_{M}^{•}X|_{∗} → X is natural in X and therefore induces a natural transformation from |Ω_{M}^{•} · |_{∗} to id. Since Ω_{M}^{•} is the nerve of a topological monoid, e is in fact a natural transformation from BΩ_{M} to id_{𝔗𝔬𝔭_{∗}}. By [Seg74, 1.5] the canonical map τ_{Ω_{M} X} : Ω_{M}X → ΩBΩ_{M}X with τ_{Ω_{M} X}(ω)(t) = (ω; 1−t, t) is a homotopy equivalence because Ω_{M}X is grouplike. The composition Ωe_{X} ∘ τ_{Ω_{M} X} : Ω_{M}X → ΩX is the map normalizing the loops to length 1 and hence a homotopy equivalence. Therefore Ωe_{X} is a homotopy equivalence. Since the maps Ω_{M}X → ΩX are natural in X, this implies the first statement of Proposition 5.1. Let M be a well-pointed grouplike monoid. Using the adjunction of the based realization and the based singular complex functors, we obtain a sequence

\xymatrix{BM = |M^{•}|_{∗} \ar[r]_{|τ_{∗,M} |_{∗}} & |S_{∗}BM|_{∗} \ar[r]_{η_{∗,BM}} & |M^{•}|_{∗} = BM}

The map η_{∗,BM} ∘ |τ_{∗,M^{•}}|_{∗} is the identity. S_{∗}BM(1) is precisely the nonassociative loop space ΩBM and, by [Seg74, 1.5], the map τ_{∗,M^{•}} is a homotopy equivalence on the 1-simplices. Furthermore S_{∗}BM(n) is based homotopy equivalent to (Ω_{M}BM)^{n} and S_{∗}BM(n) is special, i.e. it satisfies the conditions of [Seg74, 1.5]. Therefore τ_{∗,M^{•}} is a based homotopy equivalence in each dimension and thus |τ_{∗,M^{•}}|_{∗} and η_{∗,BM}. Since |a_{BM}|_{∗} is a based homotopy equivalence this implies the second statement of Proposition 5.1.

References

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Michael Brinkmeier

Fachbereich Mathematik/Informatik

Universität Osnabrück 49069 Osnabrück, Germany mbrinkme@mathematik.Uni-Osnabrueck.DE