Synopsis

Introduction
1
G.Segal's approximation theorem asserts that C ( R n , X ) is weakly homotopy equivalent to Ω n Σ n X where C ( R n , X ) denote the configuration space of finite points in R n with labels in X . In this paper, we give a definition of the configuration space with labels in a partial abelian monoid and propose a conjecture which generalizes the Segal's approximation theorem. For this purpose, the first three sections are devoted to investigate the properties of partial abelian monoids from the homotopy theoretical point of view. More precisely, the definition of partial abelian monoid is given in §2.  In §3, we give show that many functors defined on the category of topological spaces can be extended to the category of partial abelian monoids.  Such functors include the variants of the loop space functors and the classifying space functor. In §4, we settle the notation for the configuration space with labels in a partial abelian monoid and investigate some of its properties. We also state a conjecture which is the generalization of the Segal's theorem to partial abelian monoid case and indicate a direction of the solution.
The category of partial abelian monoids
2
The notion of partial monoid appeared in a work of G.Segal seems the first use of such a notion in algebraic topology. 6) Later, more general definition of such a notion was given by Shimakawa.  7) In this section, we introduce one form of such a notion adapted to the purpose of this paper. We restrict ourselves to the abelian case. Remark that our definition of partial abelian monoid is a special case of that given by Shimakawa. Definition 2.1. A space M is a (topological) partial abelian monoid if there exist a subspace M 2 ⊂ M × M and a map m : M 2 → M such that (1) M V M ⊂ M 2 , (2) (a,b) ∈ M 2 if and only if (b,a) ∈ M 2 , (3)  (m(a,b),c) ∈ M 2 if and only if (a,m(b,c)) ∈ M 2 , (4) m(a,*m) = m(*m,a) = a, (5) m(a,b) = m(b,a), and (6) m(m(a,b),c) = m(a,m(b,c)). We write m(a,b) = a + b. An element in M 2 is called a summable pair. Let Mk be a subset of Mk which consists of those k -tuples ( a 1 , . . . , a k ) such that a 1 + . . . + a k is defined. Definition 2.2. A map f : M → N between partial abelian monoids M, N is called partial abelian monoid homomorphism if f × f maps M 2 into N 2 and f ( a + b ) = f ( a ) + f ( b ). We denote the category of partial abelian monoids and partial abelian monoid homomorphisms by PAM. Example 2.3. Any space X is regarded as a partial abelian monoid by a subspace X 2 = X V X ⊂ X 2 and the folding map. We call this structure a trivial partial abelian monoid structure on X. Example 2.4. Let M be an abelian monoid and N be a subspace of M . Then N has a partial abelian monoid structure if we put N 2 = {(a,b)|a + b ∈ N }. Example 2.5. Contrary to the above example, we restrict the sum itself to give a new partial abelian monoid from an abelian monoid.  More precisely, let M be an abelian monoid.  Let N be a space homeomorphic to M and N 2 be a subspace of N × N which satisfies the conditions (1)～ (3) of Definition 2.1.
Extension of functors defined on Top x
3
In this section, we show that many functors defined on Top ∗ can be extended to functors on PAM . Results are stated as proposition, but the proof of functoriality is easy and left to the reader.
Reduced suspension functor Σ
3.1
Let Σ M denote the reduced suspension space Σ M = S 1 ∧ M. Let s ∧ a and t ∧ b be elements of Σ M . We say that s ∧ α and t ∧ b is summable if s = t and α and b are summable as elements in M . The sum is given by s ∧ a + s ∧ b = s ∧ (a + b). Proposition 3.1. The suspension functor ( − ∧ − ) is a self-functor on PAM. Suspension functor can be viewed as an application of the bivariant functor ( − ∧ − ); ΣΜ ≈ S 1 ∧ M where S 1 is considered as a partial abelian monoid with the trivial partial abelian monoid structure. 3.2 Smash product (— ∧ —) As a space, smash product M ∧ N of M and N is the quotient space M × N/M V N where M V N is identified with {( a, 0)} ∪ {(0, b )} ⊂ M × N . A class represented by ( a, b ) is denoted by a ∧ b ∈ M ∧ N , where α ∈ M and b ∈ N . We give a partial abelian monoid structure on M ∧ N by considering α ∧ b and c ∧ d are summable if one of the following holds : (1) b = d and (a,c) ∈ M 2 , or (2) a = c and (b,d) ∈ N 2 . The sum of such elements is given by
if b = d and (a, c) ∈ M 2 (a + c) ∧ b = (a + c) ∧ d if a = c and (b,d) ∈ N 2 . a ∧ (b + d) = c ∧ (b + d) Proposition 3.2. The smash product functor (− ∧ −) is a functor from the product category PAM × PAM to PAM. It is covariant on both variables.
Mapping space Map*(-,-)
3.3
As a space, the mapping space Map s ( M, N ) of M and N is the space of based maps from M to N . We give a partial abelian monoid structure on Map x ( M , N ) by considering f and g in Map x ( M , N ) is summable if f ( a ) and g ( a ) are summable in N for all α ∈ M . The sum of such maps is given by defining the map f + g by the map (f + g)(a) = f(a) + g(a). Proposition 3.3.  The mapping space functor Map ∗ ( − , − ) is a functor from the product category PAM × PAM to PAM. It is contravariant on the first variable and covariant on the second variable. Loop space functor and its variants 3.4 Let Ω M = Map ∗ ( S 1 , M ) be the space of based loops on M , thinking of M as a space. Partial abelian monoid structure of M induces a partial abelian monoid structure on Ω M = Map x ( S 1 , M ). Ω M = Map ∗ ( S 1 , M ) is a partial abelian monoid by putting (ΩΜ)2 = {(λ, μ) | (λ(t), μ(t)) ∈ M2 for all t ∈ S 1 }
Proposition 3.4. Ω is a self-functor on PAM. It is useful to introduce Moore loop space when we want a strict monoid rather than a partial monoid on the loop space. Let Ω s ( M ) denote the space of loops of length s, that is, We give a partial abelian monoid structure on Ω s ( M ) by putting (Ω s (M)) 2 = {(l 1 ,l 2 ) | (l 1 (t),l 2 (t)) ∈ M 2 for all t ∈ [0 ,s]} and
Proposition 3.5. Ω s is a self-functor on PAM. The Moore loop space is defined by Λ Μ = ∪ s > 0 Ω s ( M ) × { s } . We can identify Ω s ( M ) as a subspace of Map([0, ∞ ), M ) by extending l : [0, s] → M to [0, ∞ ) by l(t) = * for any t ∈ (s, ∞ ). We topologize AM as a subspace of Map([0, ∞), M) × [0, ∞) by this identification. AM also have a structure of partial abelian monoid . Let (α,s) and (β,t) be elements of ΛΜ, where α : [0, s] → M and β : [0, t] → M . (α, s) and (β, t) are summable if s = t and α and β are summable as elements in Ω s M . Proposition 3.6. A is a self-functor on PAM. 3.5 Classifying space functor B For any partial abelian monoid M we have a simplicial space M ∗ given by M 0 = *, M 1 = M and M k = (the space of composable k -tuples) for k ≥ 2. The structure maps of the simplicial space is s i : M k → M k +1 (0 ≤ i ≤ k ) given by s i (a 1 , ･ ･ ･ , a k ) = (a 1 , ･ ･ ･ , a i , 0 , a i +1 , ･ ･ ･ , a k ) and d i : M k → M k − 1 (0 ≤ i ≤ k ) given by
and
The classifying space of M , denoted by BM , is defined as the geometric realization of the above simplicial space. Suppose [c, m], [c', m'] ∈ BM and ( m ( t ), m' ( t )) ∈ M2 for all t ∈ c ∩ c' . We define a map m ∪ m ′ : c ∪ c ′ → M by
Then we [c, m] and [c', m'] are summable if the set {(m ∪ m')(t) | t ∈ c ∪ c'} is a summable set. (We say that a finite subset A of a partial abelian monoid M is a summable set if ( a 1 , cdots, a k ) is a summable k-tuple, where # A = k and α 1 , ･ ･ · , α k are distinct elements of A . This definition does not depend on the choice of the ordering of a 1 , ･ ･ ･ , a k since our partial monoids are abelian.) Thus the partial abelian monoid structure on BM is given by
Suppose that ([c, m], [c', m']) ∈ (BM) 2 . Then we put

Proposition 3.7. B is a self-functor on PAM.
When M is a monoid, ΣΜ is identified with the 1-skelton of BM . This also applies to our situation. Furthermore ΣΜ is a sub-partial abelian monoid of BM . We have an inclusion ι : ΣΜ → BM defined by the correspondence t ∧ a → [ t, a ], where t ∈ [0, 1] and a ∈ M . If t ∧ a and u ∧ b are summable in ΣΜ , we have t = u and ( a, b ) ∈ M 2 . Then ( ι ( t ∧ a ), ι ( t ∧ b )) = ([ t, a ], [ t, b ]) ∈ ( BM ) 2 , which shows that ( ι × ι)( M 2 ) ⊂ ( BM ) 2 . For the same pair ( t ∧ α, t ∧ b ), we have ι(t ∧ α + t ∧ b ) = ι(t ∧ (α + b )) = [t,α + b ]. On the other hand ι(t ∧ α) + ι(t ∧ b) = [t,α] + [t, b] = [t,α + b] , which shows that ι is a partial abelian monoid homomorphism.
Configuration space with summable labels
4
Salvatore considered the configuration space with labels in a non-commutative partial monoid. 5) Using a Fulton-MacPherson operad action, he showed that n -monoid is equivalent to an n -fold loop space. On the other hand, Shimakawa introduced the configuration space with labels in a partial abelian monoid and showed that if we take the infinite limit of the domain space R n , this leads to a several homology theories after taking the homotopy groups.ﾌ). The notion of the configuration space with summable labels are given by Kallel. ¹),²),³) The definition given below is a quite special case of the preceding models, but it is very simple and is sufficient for our purpose. Especially we restrict ourselves to the two-generated and abelian case. Definition 4.1. Let Z ( k ) ( M ) denote the subspace of ( R n × M ) k given by
Then we define a space C ( R n , M ) as C ( R n II Z i k 7 ( M ))/ ~, where ~ denotes the least equiva- lence relation which satisfies (R1)～(R3) below. (R1) If a i = * M then
(R2) For any permutation σ ∈ Σ k ,
(R3)  If v1 = v2 = v, ((v1,a1), · · · , (vk,ak)) ~ ((v,a1 + a2), (v3,a3), · · · , (vk,ak)). We regard C ( R n , M ) as a partial abelian monoid as follows. Let C ( R n , M ) 2 ⊂ C ( R n , M ) 2 be the subspace which consists of pairs (ξ,η) such that ( a i ,b i ) ∈ M 2 for every i where ξ = [(v1,a1),··· , (vk,ak)] and η = [(v1,b1),··· , (vk,bk)]. Then m : C ( R n , M ) 2 → C ( R n , M ) is defined by setting m(ξ,η) = [(v1,α1 + b1), · · · ,(vk,αk + bk)]. Proposition 4.2. The functor C ( R n , − ) is a functor from PAM to itself. Example 4.3. Recall from Example 2.3 that any space X has a partial abelian monoid structure called the trivial partial sum given by a subspace X 2 = X V X ⊂ X 2 and the folding map. Then C ( R n , X ) is nothing but the configuration space of finite points in R n labeled by X . Example 4.4. Let M = S 0 = {0,1} with the trivial partial abelian monoid structure. Then C ( R n , S 0 ) is homeomorphic to the configuration space of particles in R n . Example 4.5. Let M = {−1,0,1} with 1+(−1) = 0 as the only non-trivial partial sum. Then C(Rn, M) is homeomorphic to the configuration space of positive and negative particles in R n . Lemma 4.6 below says that C ( R n , − ) is a homotopy functor in some sense. Let M be a partial abelian monoid. We regard M × I as a partial abelian monoid by setting
and µm×1(m,t;n,t) = (µm(m,n),t). Homomorphisms f,g : M → N between partial abelian monoids are called homotopic via partial abelian monoid homomorphisms if there exists a homomorphism H : M × I → N such that H 0 = f, H 1 = g. Lemma 4.6. If f,g : M → N are homotopic via partial abelian monoid homomorphisms then C ( R n ,f) and C(R n , g) are homotopic via partial abelian monoid homomorphisms. Proof. Let H : M × I → N be a homotopy between f and g . Observe that we have a homomorphism
defined by the correspondence
Consider the composite
Then
Thus we see that this composition is a homotopy between C(Rn,f) and C(Rn,g) via partial abelian monoid homomorphisms. ● Proposition 4.7. BC(R n , M) is homeomorphic to C(R n , BM). Proof.  Note that any element of BC ( R n , M ) can be considered as a configuration of points in [0, 1] labelled by points in C ( R n , M ). So we may assume that the representative is of the form ( t 1 , ξ 1 ; ･ ･ · ; t k , ξ k ) where t 1 , · · · , t k are real numbers such that 0 ≤ t 1 ≤ · · · ≤ t k ≤ 1 and ξ i are elements of C ( R n , M ) A map ϕ : BC(R n , M) → C(R n , BM) is defined by the correspondence
where 0 ≤ t 1 ≤ t 2 ≤ · · ≤ t k ≤ 1,
and v is ∈ R n , α is ∈ M . We define a map ψ : C ( R n , BM ) → BC ( R n , M ) by the correspondence where β i = [ t i1 , α i1 ; · · · ; t is i , α is i ]. W
which shows that ψ ◦ ϕ is the identity on BC ( R n , M ). On the other hand we have which shows that φ ◦ ψ is the identity on C ( R n , BM )
This is a generalization of a proposition given by Segal 6) in the sense that if M = X is a space with trivial partial abelian monoid structure then BM = ΣΧ and the homeomorphism specializes to BC(Rn, X) ~ C(Rn, EX). By the above proposition, we know that B and C ( R n , − ), considered as self-functors on PAM , are commutative. We can see that C ( R n , − ) behaves well among other self-functors introduced in §2. We consider the subspace SC ( R n , M ) ⊂ BC ( R n , M ) consisting of all the elements [ t 1 , ξ 1 ; . . ; t k , ξ k ] where ξ i = [ v i 1 , α i 1 , . . . , v i r i , α i r i ] is an element of C ( R n , M ) such that v ij are distinct. Then by a careful check of the proof of Proposition 4.7 , we have the following corollary. Corolary 4.8. SC(Rn, M) is homeomorphic to C(Rn, EM). The relation between C ( R n , − ) and loop space functors can be stated as follows. Proposition 4.9.  There exist natural transformations of functors
and
Proof.  We give φ́M : C ( R n , ΩM ) → ΩC ( R n [(v1,l1), · · · , (vk,lk)] → (t → [(v1,l1(t)), · · · , (vk,lk(t))]) where v i ∈ R n and l i ∈ Ω M . Next, we show that the following diagram commutes:
where f : M → N is a partial abelian monoid homomorphism.  Observe that the image of [( v 1 , l 1 ), ··· , ( v k , l k )] E C ( R n , Ω M ) by C ( R n , f ) ◦ φ M is a loop in C ( R n , N ) given by
and the image of the same element by φ N ◦ C ( R n , f ) is given by a loop
which coincide with each other. This shows that there exists a natural transformation −−−− The proof applies when Ω is replaced by Ω s .
Proposition 4.10. There exists a natural transformation of functors
Proof. In this case, we give φ M : C ( R n , ΛΜ ) → Λ C ( R n , M ) by φ M ([( v 1 , ( l 1 , s 1 )), . . . , ( v k , ( l k , s k )]) = (l, s) where s = Max{ s 1 , · · · , s k } and l : [0, s] → C ( R n , M ) is given by l ( t ) = [( v 1 , l 1 ( t )), · · · , ( v k , l k ( t ))]. Here, the domain of l i (t) is extended to [0, s] by considering l i (t) = * for t > s i . Proof that φ gives the natural transformation is similar to the proof of the above proposition. ● We may ask the homotopy type of C ( R n , M ). We expect that a generalization of the Segal's approxi- mation theorem 6) holds. In the Segal's theorem, it was assumed that X is path-connected. This theorem is well understood in the context of group completion if we say the connectivity condition as 'π 0 ( X ) is a group.' When we attempt to generalize the Segal's theorem to partial abelian monoid M in place of X , we need the corresponding notion. The following notion is one such generalization of the connectivity condition. Definition 4.11. A partial abelian monoid is grouplike if for any a ∈ M there exists b ∈ M such that a + b ∈ M 0 , where M 0 denotes the path-component of 0. If we think of the Segal's theorem as a theorem about the self-functors C ( R n , − ) (in Segal's nota- tion, C n (−)) and Ω n Σ n on the category of based spaces, then the generalization to the partial abelian monoid case should be the theorem about the self-functors on PAM . We propose the following conjec- ture. Conjecture 4.12. C(Rn, M) is homotopy equivalent to Ωn Bn M if M is grouplike. By Proposition 4.7, we may expect that we can generalize the lines of the Segal's paper 6) to prove the above conjecture. But we do not know how to manage the term 'weakly homotopy equivalent as partial abelian monoid ' at this point. For example suppose we define this notion as follows : Predefinition 4.13. Partial abelian monoids M and N are weakly homotopy equivalent if there exists a partial abelian monoid homomorphism M → N which is a weak homotopy equivalence i.e. there exists an inverse homomorphism of f* : π*(M) → π*(N). Then we do not know how to show the statement such as : Preproposition 4.14. C(Rn,M) is weakly homotopy equivalent to C(Rn,N) whenever M is weakly homotopy equivalent to N . It seems that we need a more sophisticated definition of the homotopy groups of partial abelian monoids or homomorphisms between them so that they reflects the partial abelian monoid structure to ensure that such a statement holds. We propose the other problems related to the above problem. Problem 4.15. Encode the partial abelian monoid structure properly in the homotopy group π ⋆ ( M ). It seems that Definition 4.11 can be stated in terms of π 0 ( M ) if we answer to this problem. The following problem is a very natural question, which is related to the Preproposition 4.14. Problem 4.16. Describe the homotopy group π ∗ (C(R n , M)) in terms of π ∗ (M).