Documenta Math. 613 Strongly Homotopy-Commutative Monoids Revisited

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.


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.

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.
A continuous homomorphism F : The 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 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 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 : 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.
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 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 The product of two paths (ω, l) and (ν, k) is given by (ρ, l + k), with

Documenta Mathematica 5 (2000) 613-624
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 TM : W M → P (EW M ; e, W M ) such that the following diagram commutes up to homotopy through homomorphisms.
x x q q q q q q q q q q q W Ω M BW M For each well-pointed space X, we chose E X to be the dotted arrow in the following diagram.
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 : 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 and since e BW M is a based homotopy equivalence by Proposition 5.1 this results in and the naturality of several maps leads to Since ε ΩM X and Ω M e X are homotopy equivalences the homomorphisms W Ω M e X and W Ω B ε ΩM X represent isomorphisms in HGr H . Therefore we have The facts that T ΩM X is an isomorphism in HGr H and that Documenta Mathematica 5 (2000) 613-624 3 Hopf-objects Definition 3.1.An H-or Hopf-object (X, µ, ρ) in a monoidal category1 (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.
o o y y t t t t t t t t t t X.

A morphism of H-objects (or
commute for each X ∈ C and Y ∈ D, then there exists an adjoint pair of functors Hopf F : Hopf HC Hopf D : Hopf G. Proof.Hopf F is given by Hopf F (X, µ, ρ) = (F X, F µ • ϕ, F ρ) and Hopf F (f ) = F f, with ϕ : F X ⊗F X → F (X X) the natural transformation.Its adjoint Hopf G 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.
This construction is compatible with the composition and we can define a functor ⊗ : 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.
, (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.

Documenta Mathematica 5 (2000) 613-624
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 CWcomplexes, 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.

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 ∆ 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 ∇ * : ∆ → Top * .We define the based geometric realization of a based simplicial space X as 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 (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 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 ΩM X : Ω M X → ΩBΩ M X with τ ΩM X (ω)(t) = (ω; 1−t, t) is a homotopy equivalence because Ω M X is grouplike.The composition Ωe X • τ ΩM 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 TM 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.W M TM / / TM W Ω M BW M εΩ M BW M P (EW M ; e, W M ) P (pM ) / / 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.
The H-objects of C and the H-morphisms form a category Hopf C. 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 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.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

εM ×εN w w p
p p p p p p p p p p 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.

Theorem 4. 2 .⊂
The full subcategories Hopf HGr CW H Hopf HGr H of Hobjects of the homotopy type of CW-complexes, and Hopf Top * ,CW H ⊂ Hopf Top * H of connected H-spaces of the homotopy type of CW -complexes, are equivalent.
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 : ∆ 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) Documenta Mathematica 5 (2000) 613-624and 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.