Configuration spaces with summable labels
Paolo Salvatore *
July 12, 1999
Abstract
Let M be an n-manifold, and let A be a space with a partial sum
behaving as an n-fold loop sum. We define the space C(M; A) of configu-
rations in M with summable labels in A via operad theory. Some examples
are symmetric products, labelled configuration spaces, and spaces of ra-
tional curves. We show that C(In; @In; A) is an n-fold classifying space
of C(In; A), and for n = 1 it is homeomorphic to the classifying space by
Stasheff. If M is compact, parallelizable, and A is path connected, then
C(M; A) is homotopic to the mapping space Map(M; C(In; @In; A)).
Introduction
The interest in labelled configuration spaces in homotopy theory dates back
to the seventies. May [15] and Segal [20] showed that the `electric field map'
C(Rn; X) ! nn(X) is a weak homotopy equivalence if X is path connected,
and in general is the group completion. Segal showed later [21] that the inclu-
sion Rat*(S2) ,! 2S2 of the space of based rational selfmaps of the spheres
into all based selfmaps is the group completion. He used the identification of
Rat*(S2) with a space of configurations in C with partially summable labels
in N _ N, by counting zeros and roots multiplicities. Guest has recently ex-
tended his framework in [9] to the space of based rational curves on projective
toric varieties. Labelled configuration spaces on manifolds have been studied
by B"odigheimer in [3], where the labels are in a based space, and by Kallel in
[10], where the summable labels belong to a discrete partial abelian monoid. In
both cases the authors have theorems of equivalence between configuration and
mapping spaces. We define configuration spaces on a manifold M with labels
in A, where A need not to be abelian. It is sufficient that A has a partial sum
that is homotopy commutative up to level dim(M). The definition is not trivial
and involves tensor products over the Fulton-MacPherson operad. A substantial
part of the paper introduces the necessary tools. We generalize the results lis*
*ted
above to the non-abelian setting, and construct a geometric n-fold delooping in
one step. Here is a plan of the paper:
____________________________*
Supported by the Marie-Curie Research Training Grant ERBFMBICT961611
1
In the first section we define the preliminary notion of a partial algebra o*
*ver
an operad and its completion. In the second section we introduce the Fulton-
MacPherson operad Fn. We call an algebra over Fn an n-monoid. A 1-monoid
is exactly an A1 -space [12]. In the third section we describe the homotopical
algebra of topological operads and their algebras. The main results characterize
the homotopy type of Fn.
Corollary 3.8. The unbased Fulton-MacPherson operad F"nis cofibrant.
Proposition 3.9. The operad of little n-cubes is weakly equivalent to Fn.
This implies that the structure of n-monoid is invariant under based homo-
topy equivalences, and any connected n-monoid has the weak homotopy type of
a n-fold loop space. In the fourth section we recall from [14] that a partial c*
*om-
pactification C(M) of the ordered configuration space on an open parallelizable
n-manifold M is a right module over Fn. We define the configuration space
C(M; A) on M with summable labels in a partial n-monoid A, by tensoring
C(M) and A over the operad Fn.
The definition of C(M; A) is extended to a general open n-manifold M when
A is framed, in the sense that A has a suitable GL(n)-action. In the fifth sect*
*ion
we define C(M; N; A) for a relative manifold (M; N) by ignoring the particles
in N. For n = 1 we obtain the well known construction by Stasheff:
Proposition 5.11. If A is a 1-monoid, then C(I; @I; A) is homeomorphic
to the classifying space B(A) by Stasheff.
The n-monoid completion of a partial n-monoid A is C(In; A), up to homo-
topy. We obtain its n-fold delooping in one step.
Theorem 6.3. If A is framed, then the group completion of C(In; A) is
nC(In; @In; A).
Finally we characterize configuration spaces on manifolds under some con-
ditions.
Theorem 6.6. If M is a compact closed parallelizable n-manifold and A
is a path connected partial framed n-monoid, then there is a weak equivalence
C(M; A) ' Map(M; Bn(A)).
As corollary we obtain a model for the free loop space on a suspension built
out of cyclohedra. This answers a question by Stasheff [22].
Corollary 6.7. If X is path connected and well pointed, then there is a
weak homotopy equivalence C(S1; X) ' Map(S1; X).
2
The author is grateful to M. Markl and J. Stasheff for many valuable sug-
gestions.
1 Partial modules over operads
Let C be a symmetric closed monoidal category, with tensor product and unit
element e. We assume that C has small limits and colimits.
Definition 1.1.A -object X in C is a collection of objects X(n), for n 2 N,
such that X(n) is equipped with an action of the symmetric group n.
The category of -objects in C will be denoted by C. We observe as in [8]
that C is a monoidal category as follows: given two objects A and B, their
tensor product A B is defined by
1a a Ok
(A B)(n) = A(k) k ( Bss-1(i));
k=0 ss2Map(n;k)i=1
where Map(n; k) is the set of maps from {1; : :;:n} to {1; : :;:k}. Here each
element BS, where S is a set of numbers, is identified to B#S by the order
preserving bijection, and the action of n is given accordingly. There is a
natural embedding functor j : C ,! C considering an object X as a -object
concentrated in degree 0, so that
(
j(X)(n) = X if n = 0;
; if n 6=:0
with the trivial actions of the symmetric groups. Here ; denotes the initial
object of C. The functor j is left adjoint to the forgetful functor B 7! B(0) f*
*rom
C to C. More generally we have an embedding functor jn : nC ! C, that is
left adjoint to the forgetful functor B 7! B(n).
The unit element of C is defined by
(
(n) = e if n = 1;
; if n 6=:1
Definition 1.2.[8] An operad in C is a monoid in the monoidal category C.
We denote the category of operads in C by OP (C).
This means that an operad (F; ; j) is a -object F together with compo-
sition morphism : F F ! F and unit morphism j : ! F , such that the
associativity property ( F ) = (F ) : F F F ! F and the unit
property (F ) = ( F ) = idF : F ! F hold. Note that the functor
F _ forms a triple.
3
Example 1.3. The category CH R of non negatively graded chain complexes
over a commutative ring R is monoidal by the tensor product. The operads in
CH R are called differential graded operads over R.
Definition 1.4.[14] Given an operad F in C, a left F -module A is a -object
A, with a morphism ae : F A ! A of -objects such that
ae(F ae) = ae( A) : F F A ! A and ae(j A) = idA : A ! A.
In other words A is an algebra over the triple F _. Dually we define the
notion of right F -module. We denote the category of left F -modules by ModF ,
and the category of right F -modules by FMod.
Definition 1.5.If F and G are operads in C, then a F -G-bimodule A is a left
F -module and a right G-module such that the left F -module structure map is a
right G-homomorphism.
Definition 1.6.An algebra X over an operad F , or F -algebra, is an object X
of C together with a left F -module structure on j(X).
We denote the category of F -algebras by AlgF . Moreover we will denote by
F (Y ) the free F -algebra generated by the object Y in C. This object is defin*
*ed
by j(F (Y )) = F j(Y ).
Definition 1.7.A partial left F -module A is a -object A in C together with a
monomorphism i : Comp ,! F A in C and a composition map ae : Comp ! A
such that
1.The unit j A : A ! F A factors uniquely through "j: A ! Comp and
the composition ae("j) = idA is the identity.
2.The pullbacks in C of i : Comp ,! F A along the two maps (A)(F
i) : F Comp ! F A and F ae : F Comp ! F A coincide. Moreover
the compositions of the two pullback maps with ae : Comp ! A coincide.
Partial right F -modules are defined dually. A partial F -algebra is an ob-
ject X of C such that j(X) is a partial left F -module. We denote the cate-
gories of partial left F -modules, right F -modules and F -algebras respectively
by P artModF ; FP artMod, and P artAlgF .
A morphism g : (A; CompA ) ! (B; CompB ) of partial left F -modules is
a morphism in C such that (F g)iA : CompA ! F B factors through
g": CompA ! CompB , and gaeA = aeB "g.
We exhibit a functor from partial to total left modules that is left adjoint*
* to
the forgetful functor. The analogous construction for right modules is exactly
dual.
If A is a partial F -left module, then define A^ by the coequalizer in the
category C
(A)(Fi)-
F Comp __________-F A ........-A^:
Fae
4
Proposition 1.8. There is a left F -module structure on ^A.
Proof. The proof is modelled on Lemma 1.15 in [8]. The coequalizer above is
reflexive because the input arrows admit the common section F "j: F A !
F Comp. Now ^Aadmits the structure of left F -module, because by Lemma
2.3.8 in [17] F _preserves reflexive coequalizers. Moreover ^Ais the coequaliz*
*er_
of the pair above in the category of left F -modules. |__|
Proposition 1.9. The completion A 7! ^Ainduces a functor that is left adjoint
to the forgetful functor U : ModF ! P artModF .
Definition 1.10. For any right F -module C with structure map oe : C F ! C
and a partial left F -module (A; Comp; i) we define the tensor product C F A
as coequalizer in C
(oeA)(Ci)-
C CompA __________-C A ........-C F A :
Cae
Dually we define the tensor product of a partial right F -module and a left
F -module.
Proposition 1.11. Given a partial right F -module A, an F -G-bimodule B,
and a partial left G-module C, there are natural isomorphisms
(A F B) G C ~=A F (B G C):
The isomorphism holds because the tensor product is a left adjoint and
preserves colimits.
2 The Fulton-MacPherson operad
The category CG of compactly generated weak Hausdorff topological spaces
is a closed monoidal category with all limits and colimits, hence it satisfies
the assumptions of the previous section. We note however that in general the
forgetful functor to the category of sets does not preserve colimits. Operads
and modules in CG shall be called simply topological operads and topological
modules.
The key topological operads in this paper are the Fulton-MacPherson op-
erads, that are suitable cofibrant versions of the little cubes operads. They
were introduced in [8]. We recall their definition. Consider the differential-
geometric blow-up of (Rn)k along the small diagonal = {x1; : :;:xk | x1 =
. .=.xk}. The blow-up is explicitly obtained if we replace the diagonal by its
normal sphere bundle.P The fiber of the trivial normal bundle at the origin is
F = {y1; : :;:yk | i=ki=1yi= 0} and the sphere bundle P F = (F - 0)=(R+ ) can
be seen as the space of closed half-lines in F . Then the blow-up is
Bl ((Rn)k) = {(x; y) 2 (Rn)k x P F | x - ss (x) 2 y};
5
P i=k P i=k
where the orthogonal projection is ss (x1; : :;:xk) = ( i=1xi=k; : :;: i=1xi*
*=k):
For any set S {1; : :;:k} let us denote by Bl ((Rn)S) the blow-up of (Rn)S
along its small diagonal.
Let C0k(Rn) Map({1; : :;:k}; Rn) be the space of ordered pairwise distinct
k-tuples in Rn. There is a natural right k-action on this space, and we consider
it as left k-space by the opposite action. As C0k(Rn) does not intersect any
diagonal, there is a natural embedding
Y
j : C0k(Rn) ! Bl ((Rn)S):
S{1;:::;n}; #S2
Definition 2.1.The Fulton-MacPherson configuration space Ck(Rn) is the clo-
sure of the image of j.
We note that GL(n) acts diagonally on each blowup, j is a GL(n)-equivariant
map and therefore Ck(Rn) is a GL(n)-space.
In a similar way we define the Fulton-MacPherson configuration space Ck(M)
of a smooth open manifold M. In this case one builds the differential-geometric
blowups of Mk along the diagonal M by gluing together Mk - M and the
normal sphere bundle via a tubular neighbourhood of M in Mk. It turns out
that Ck(M) is a manifold with corners k-equivariantly homotopy equivalent to
its interior, the ordinary configuration space C0k(M) of ordered pairwise disti*
*nct
k-tuples in M.
There is a blow-down map b : Ck(M) ! Mk such that the composite
C0k(M) ____j-Ck(M) ____b-Mk is the inclusion. We will say that the blow-
down map gives the macroscopic locations of the particles.
There is a characterization of the Fulton-MacPherson configuration space by
means of trees due to Kontsevich. For us a tree is an oriented finite connected
graph with no cycles such that each vertex has exactly one outcoming edge. An
ordered tree is a tree together with an ordering of the incoming edges of each
vertex. The ordering is equivalent to the assignation of a planar embedding.
The only edge with no end vertex is the root, the edges with no initial vertex
are the twigs, and all other edges are internal. A tree on a set I is a tree to*
*gether
with a bijection from the set of its twigs to I. The valence of a vertex is the
number of incoming edges. Let G(n) be the group of affine transformations of
Rn generated by translations and positive dilatations.
Proposition 2.2. [12] Let M be an open manifold. Then each element in
Ck(M) is uniquely determined by:
1.Distinct macroscopic locations P1; : :;:Pl2 M, with 1 l k.
P l
2.For each 1 i l a tree Ti with fi twigs, so that i=1fi = k, and for
each vertex in Ti of valence m an element in C0m(oPi(M))=G(n), where
oPi(M) is the tangent plane at Pi.
3.A global ordering of the k twigs of the trees.
6
Definition 2.3.If b : Ck(Rn) ! (Rn)k is the blowdown map, then the Fulton-
MacPherson space is Fn(k) = b-1({0}k).
This space contains all configurations macroscopically located at the origin.
Proposition 2.4. [8] The space Fn(k) is a manifold with corners, and it is a
compactification of C0k(Rn)=G(n).
The faces of Fn(k) are indexed by trees on {1; : :;:k}, and the codimension
of a face is equal to the number of internal edges of the indexing tree.
Proposition 2.5. [14] The spaces Fn(k) k 0 assemble to form a topological
operad. Moreover Fn(k) is a k-equivariant deformation retract of Ck(Rn).
The composition law is easily described in terms of trees: each element in
Fn is described by a single tree by Proposition 2.2. If a 2 Fn(k) and bj 2 Fn(i*
*j)
for j = 1; : :;:k then a O (b1; : :;:bk) 2 Fn(i1 + . .+.ik) corresponds to the *
*tree
obtained by merging the j-th twig of the tree of a with the root of the tree of*
* bj
for j = 1; : :;:k, and assigning the new twigs the induced order. This operation
on trees will be called grafting. Note that Fn(1) is a point, the unit of the
operad, and is represented by the trivial tree. We assume that Fn(0) is a point,
the empty configuration. We stress the fact that Getzler and Jones in [8] consi*
*der
the unpointed version "Fnsuch that "Fn(k) = Fn(k) for k > 0 and "Fn(0) = ;.
Their paper focuses on the differential graded operad en = H*(F"n; Q), the
rational homology of F"n. They denote H*(Fn; Q) by e+n. The deformation
retraction r : Ck(Rn) x I ! Ck(Rn) such that r(Ck(Rn) x {1}) = Fn(k) is
defined for t 6= 0 and x 2 C0k(Rn) by r(x; t) = xt.
Definition 2.6.We call an algebra over Fn an n-monoid, and an algebra over
F"nan n-semigroup.
Example 2.7. [12] The 1-monoids are the A1 -spaces.
In fact F1(i) = Kix i, where Ki denotes the associahedron by Stasheff
[22], so F1 is the symmetric operad generated by the non-symmetric Stasheff
operad K. But an A1 space is by definition an algebra over K.
3 Homotopical algebra and the little discs
We describe the closed model category structure of the categories of topological
operads and their algebras.
Definition 3.1.[6] A cofibrantly generated model category is a closed model
category [16], together with a set I of generating cofibrations, and a set J of
generating trivial cofibrations, so that the fibrations and the trivial fibrati*
*ons
are respectively the maps satisfying the right lifting property with respect to*
* the
maps in J and I.
7
Consider the free operad functor T : (CG ) ! OP (CG ), left adjoint to the
forgetful functor U : OP (CG ) ! (CG ). Let Sn be the family of subgroups of
n. The simplicial version of the following proposition is 3.2.11 in [17].
Proposition 3.2. [18] The category of topological operads is a cofibrantly gen-
erated model category, with the following structure:
1.The set of generating cofibrations is I = {T(@Ii x H \ n ,! Ii x H \
n) | i; n 2 N; H 2 Sn };
2.The set of generating trivial cofibrations is
J = {T((Ii-1 x {0}) x H \ n ,! Iix H \ n) | i; n 2 N; H 2 Sn };
3.A morphism f is respectively a weak equivalence or a fibration if for any
n 2 N and H 2 Sn the restriction fHn of fn to the H-invariant subspaces
is respectively a weak homotopy equivalence or a Serre fibration.
There is a functorial cofibrant resolution for operads, introduced in [1]. L*
*et
A be a topological operad. Let Mk be the set of isomorphism classes of ordered
trees on {1; : :;:k}, and for each tree t let V (t) be the set of its vertices *
*and
E(t) the set of its internal edges. For each vertex x 2 V (t) let |x| be its va*
*lence.
Definition 3.3.The space of ordered trees on {1; : :;:k} with vertices labelled
by elements of A, and with internal edges labelled by real numbers in [0; 1] is
a Y
Tk(A) = ( A(|x|) x [0; 1]#E(t)) :
t2Mk x2V (t)
Let Tt be the summand indexed by a tree t 2 Mk. For each internal edge
e 2 E(t) the operad composition induces a map @e : Tt ! Tt-e, where t - e
is obtained from t by collapsing e to a vertex. If e goes from x to y, |y| = n,
and e is the i - th incoming edge of y, then @e(x) is the multiplication of the
composition i: A(|x|) x A(|y|) -! A(|x| + |y| - 1) by the identity maps of the
vertices in V (t) - {x [ y}.
Definition 3.4.The space W A(k) is the quotient of Tk(A) under the following
relations:
1.Suppose that t 2 Tk(A), v is a vertex of t of valence n labelled by ff 2 A*
*(n),
the subtrees stemming from v are t1 < . .<.tn, and oe 2 n. Then t is
equivalent to the element obtained from t by replacing ff by oe-1ff and by
permuting the order of the subtrees to toe1< . .<.toen.
2.If t 2 Tk(A) has an edge e of length 0, then t is equivalent to the labell*
*ed
tree obtained by collapsing e to a vertex, and composing the labels of its
vertices.
3.If t 2 Tk(A) has a vertex w of valence 1 labelled by the unit 2 A(1) of
the operad A, then t is equivalent to the labelled tree obtained by removi*
*ng
w. If w is between two internal edges of lengths s and t, then we assign
length s + t - st to the merged edge.
8
There is an action of k on W A(k) induced by permuting the labelling of
the twigs of elements in Tk(A).
Proposition 3.5. [1] There is an operad structure on W A, defined by grafting
trees, and by assigning length 1 to the new internal edges. A natural ordering
of the twigs of the composite is induced. The trivial tree consisting of an edge
with no vertices is the identity of W A.
For us a cofibration of topological spaces is the retract of a generalized C*
*W-
inclusion [6]. We say that a pointed space (X; x0) is well-pointed if the inclu*
*sion
{x0} ,! X is a cofibration. The following proposition is essentially proved in
[1].
Proposition 3.6. Let A be a topological operad such that (A(1); ) is well-
pointed and each A(n) is a cofibrant space. Then W A is a cofibrant resolution
of A.
Proposition 3.7. There is an isomorphism of topological operads W (F"n) ~=
F"n.
Proof. We observe that W (F"n)(i) is obtained by gluing together for each face
S of Fn(i) of codimension d a copy of S x [0; 1]d. This is true because the
codimension of a face of the manifold with corners Fn(i) is equal to the number
of internal edges of the associated tree. Then W (F"n)(i) admits the structure
of manifold with corners diffeomorphic to Fn(i), and the composition maps of_
both operads are described by grafting of trees. |__|
Corollary 3.8. The operad "Fnis cofibrant.
Let Dn be the operad of little n-discs. The space Dn(k) is the space of k-
tuples of direction preserving affine selfembeddings of the unit n-disc with pa*
*ir-
wise disjointed images. The operad structure is defined by multicomposition of
the embeddings. There is a sequence of k-equivariant homotopy equivalences
Dn(k) ! C0k(In) ,! C0k(Rn) ,! Ck(Rn) ! Fn(k): The first map sends the
little discs to their centers, the last is a deformation retraction. The inclus*
*ion
C0k(In) ,! C0k(Rn) is a k-equivariant homotopy equivalence because the inclu-
sion In ,! Rn is isotopic to a homeomorphism. The image of the composite rk
is the interior of the manifold with corners Fn(k). It follows that the -map
r = {rk} is not an operad map because all elements in the boundary of Fn(k)
are composite.
Proposition 3.9. The operad Dn is weakly equivalent to Fn.
Proof. We build an extension R : W Dn ! Fn of r : Dn ! Fn that is a map of
operads and a weak equivalence.
An element a in W Dn(k) is represented by a labelled tree o 2 Tk(Dn) on
{1; : :;:k}. If the i-tuple (fv1; : :;:fvi) labels a vertex v of valence i, th*
*en for
each j we associate the embedding fvjto the j-th incoming edge ej(v) of v. The
equivalence relation defining W Dn preserves this association. We observe inci-
dentally that the multicomposition of the labels of o is the k-tuple of embeddi*
*ngs
9
(g1; : :;:gk) such that for each j gj is the composition of the embeddings asso*
*ci-
ated to the edges along the unique path from the j-th twig to the root. Suppose
that the internal edges of o are labelled by numbers in (0; 1). Let l(e) denote
the length of an edge e and if r 2 (0; 1] let ffir be the dilatation of the n-d*
*isc by
r. Consider the labelled tree o0 obtained from o by replacing for each vertex v
and for each j = 1; : :;:|v| the embedding fvjby the rescaling fvjO ffil(ej(v))*
*. Let
b be the multicomposition of the labels of o0, and set Rk(a) = rk(b). We have
defined Rk on a dense subspace of W Dn(k). The map Rk extends to W Dn(k)
and R is an operad map, because the boundary and the composition of Fn are
described by a limit procedure. Let ik : Dn(k) ! W Dn(k) be the inclusion
such that ik(a) is represented by the tree on {1; : :;:k} with a single vertex
labelled by a. The map Rk : W Dn(k) ! Fn(k) is a k-equivariant homotopy
equivalence for each k because ik is such [1] and Rkik = rk. In particular_R_is
a weak equivalence of topological operads, and Dn ' Fn. |__|
The simplicial analogue of the following proposition is 3.2.5 in [17].
Proposition 3.10. [18] [19] Let F be a topological operad. Then the category
AlgF is a cofibrantly generated model category with the following structure:
1.The set of generating cofibrations is I = {F (@Ii) ,! F (Ii) | i 2 N}.
2.The set of trivial generating cofibrations is J = {F (Ii-1x{0}) ,! F (Ii) *
*| i 2
N}.
3.A F -homomorphism is a weak equivalence or a fibration if it is respective*
*ly
a weak homotopy equivalence or a Serre fibration.
Under mild conditions there is a functorial cofibrant resolution of topologi*
*cal
algebras over operads, introduced in [1]. Let A be a topological operad. Consid*
*er
the -space W +A defined similarly as W A, except that relation 3 is not applied
if w is the root vertex. It turns out that W +A is an A-W A-bimodule, by action
of A on the label of the root, and by grafting trees representing elements of W*
* A.
Let X be an A-algebra. It has a W A-algebra structure induced by the projection
" : W A ! A. We define the A-algebra UA (X) = W +A WA X. The projection
W +A ! W A, obtained by extending relation 3 to the root vertex, induces an
A-homomorphism ss : UA (X) ! X, that is a deformation retraction, see p. 51
of [1].
Proposition 3.11. [18] If X is a cofibrant space then UA (X) is a cofibrant
A-algebra.
Definition 3.12. If A is a topological operad, X and Y are A-algebras, then a
homotopy A-morphism from X to Y is an A-homomorphism from UA (X) to Y .
Proposition 3.13. If F is a topological operad, X is the retract of a generaliz*
*ed
CW -space, and Ho(AlgF ) is the homotopy category, then Ho(AlgF )(X; Y ) =
AlgF (UF (X); Y )= '.
10
Proof. The set of right homotopy classes [X; Y ] in the sense of [16] is the se*
*t of
F -homomorphisms from a cofibrant model of X to a fibrant model of Y modulo
right homotopy. Now UF (X) is a cofibrant resolution of X, and Y is fibrant
because every object is such. It is easy to see that the the right homotopy
classes of F -homomorphisms from UF (X) to Y are ordinary homotopy classes,_
because Y Iis a path object. |__|
This result is consistent with the formulation of the homotopy category of
F -algebras in [1].
Proposition 3.14. If Z is an n-semigroup and p : Y ! Z is a homotopy
equivalence, then Y has a structure of n-semigroup such that p extends to a
homotopy "Fn-morphism.
Proof. It is sufficient to observe that W "Fnis homeomorphic to "Fn, and apply_
the homotopy invariance theorem 8.1 in [1]. |__|
4 Modules and configuration spaces with summable
labels
Proposition 4.1. [14] For any parallelizable`open manifold M of dimension n,
the space of configurations C(M) = k2NCk(M) is a right Fn-module.
Proof. We choose a trivialization of the tangent bundle o(M) ~=M x Rn. Then
the composition C(M) Fn ! C(M) is described by grafting of trees repre-__
senting elements as in 2.2. |__|
Markl in [14] gives a similar picture for generic open manifolds by introduc*
*ing
the framed Fulton-MacPherson operads.
Definition 4.2.Let G be a topological group, and let F be a topological operad
such that F (i) is a G x i-space for each i, and the structure map of F is G-
equivariant. The semidirect product F oG is the operad defined by (F oG)(i) =
F (i) x Gi, with structure map
"((x; g1; : :;:gk); (x1; g11; : :;:gm11); : :;:(xk; g1k; : :;:gmkk)) =
= ((x; g1x1; : :;:gkxk); g1g11; : :;:gkgmkk) :
Definition 4.3.The framed Fulton-MacPherson operad is the semidirect prod-
uct fFn = Fn o GL(n).
Definition 4.4.Let M be an open n-manifold. The GL(n)-bundle of frames
on M induces a GL(n)k-bundle fCk(M) on Ck(M), acted on by k, that we
call the framed configuration space of k frames in M.
Proposition 4.5. [14] The -space fC(M) of framed configurations is a right
module over fFn.
11
An element of the framed configuration space fCk(M) is uniquely deter-
mined by labelled trees as in Proposition 2.2, and by additional k frames of the
tangent planes associated to the k twigs. A smooth embedding i : M ,! N
of open n-manifolds induces a right fFn-homomorphism fC(i) : fC(M) ,!
fC(N).
Remark 4.6. If M is a Riemannian n-manifold, then we can define for each k
a O(n)k-bundle fO Ck(M) over Ck(M), so that fO C(M) is a right Fn o O(n)-
module. If M is oriented then we can define a SO(n)k-bundle fSO Ck(M) on
Ck(M) so that fSO C(M) is a right Fn o SO(n)-module.
Definition 4.7.We call an algebra over fFn a framed n-monoid.
Hence a framed n-monoid is an n-monoid equipped with an action of GL(n),
that is compatible with the n-monoid structure map.
Definition 4.8.A partial framed n-monoid is a partial n-monoid with an ac-
tion of GL(n), such that GL(n) preserves Comp and respects the partial com-
position.
Definition 4.9.Let fDn(k) be the space of k-tuples of of affine selfembeddings
of the unit n-disc that preserve angles and have pairwise disjointed images. The
multicomposition gives fDn the structure of an operad, that we call the operad
of framed little n-discs.
Remark 4.10. Consider the iterated loop space n(X; x0) as the space of maps
from the closed unit n-disc to X, sending the boundary to the base point x0. Th*
*is
space is an algebra over fDn.
Proposition 4.11. The operad of framed little n-discs fDn is weakly equivalent
to the framed Fulton-MacPherson operad fFn.
Proof. We apply the same proof of Proposition 3.9 to show that fDn ' Fn o_
O(n), and conclude by the homotopy equivalence O(n) ,! GL(n). |__|
If we restrict to the suboperad fDn fDn containing orientation preserving
embeddings, then we obtain a weak equivalence fDn ' Fn o SO(n).
Definition 4.12. Let A be a partial n-monoid, and let M be an open paral-
lelizable manifold of dimension n. Then the space of configurations in M with
partially summable labels in A is C(M; A) := C(M) Fn A.
`
An element of C(M) A = kCk(M) xk Ak consists by 2.2 of a finite set
of trees based at distinct points in M, with vertices labelled by Fn and twigs
labelled by A. The equivalence relation defining C(M; A) says that if some
twigs labelled by a1; : :;:ak are departing from a vertex labelled by c 2 Fn(k)
in t 2 C(M) A and ae(c; a1; : :;:ak) is defined, then we identify t with the
forest obtained from t by cutting such twigs, and by replacing their vertex by
a twig labelled by ae(c; a1; : :;:ak). Furthermore if the i-th twig departing f*
*rom
a vertex labelled by c in t is labelled by the base point a0 then we identify t
12
to forest obtained by cutting the twig and by replacing the label c by si(c),
where si : Fn(k) ! Fn(k - 1) is the projection induced by forgetting the i-th
coordinate.
If A is an n-monoid, then by iterated identifications any element in C(M; A)
has a unique representative consisting of a finite set of trivial trees in M, or
points, with labels in A - {a0}.
We denote by | | : CG ! Set be the forgetful functor.
Proposition 4.13. Suppose that the inclusion Comp ,! F (A) is a cofibration,
and A is well-pointed. Then
1.|C(M; A)| = |C(M)| |Fn||A| ;
2.the space C(M;`A) has the weak topology with respect to the filtration
Ck(M; A) = Im( ik C(M)ixi Ai); k 2 N.
Proof. If A is a proper n-monoid, then we have relative homeomorphisms
(Ck(M); @Ck(M)) xk (A; a0)k -! (Ck(M; A); Ck-1(M; A))
for k 1, and we conclude by 8.4, 9.2 and 9.4 in [23]. If A is a partial n-mono*
*id,
then we denote by Ri Ci(M) xi Ai the space of reducible elements that are
equivalent to an element of some Cj(M) xj Aj with j < i. For example,
R1= M x {a0};
R2= (C2(M) x2 (A _ A)) [ (M x Comp2);
R3= (C3(M) x3 (A _ A _ A)) [ (C2(M) x2 (Comp2 x A)) [ (M x Comp3):
We have relative homeomorphisms (Ci(M)xi Ai; Ri) ! (Ci(M; A); Ci-1(M; A))_;
and we argue similarly. |__|
Definition 4.14. Suppose that M is an open n-dimensional smooth manifold,
and A is a partial framed n-monoid. Then the space of configurations in M with
labels in A is C(M; A) := fC(M) fFn A.
Note that if M is parallelizable then the definition is consistent with 4.12.
In fact the framed configurations in M are given by fC(M) = C(M) Fn fFn
and by 1.11
fC(M) fFn A = C(M) Fn fFn fFn A = C(M) Fn A:
Proposition 4.15. Let A be a partial framed n-monoid with base point a0 such
that the inclusions Comp ,! fFn(A) and {a0} ,! A are cofibrations of GL(n)-
spaces. Let M be an open n-manifold. Then
1.|C(M; A)| = |fC(M)| |fFn||A| ;
2.the space C(M;`A) has the weak topology with respect to the filtration
Ck(M; A) = Im( ik fC(M)ixi Ai); k 2 N.
13
We give some examples of configuration spaces with summable labels. Let
us denote by ^Anthe completion of a partial n-monoid A.
Proposition 4.16. If A is a partial n-monoid, then there is a strong deforma-
tion retraction wA : C(Rn; A) ! ^An.
Proof. It is sufficient to observe that there is a deformation retraction of ri*
*ght
Fn-modules w : C(Rn) ! Fn. If an element x 2 C(Rn; A) is represented by a fi-
nite number of labelled trees o1; : :;:ok based at distinct points P1; : :;:Pk *
*2 Rn,
then wA (x) is represented by the single tree obtained by connecting o1; : :;:ok
to a root vertex labelled by the class [P1; : :;:Pk] 2 C0k(Rn)=G(n) _Fn(k)._
|__|
Example 4.17. If M is a discrete partial monoid, then M^1 has the homotopy
type of its monoid completion. If M is abelian then M^1 has the homotopy type
of its abelian monoid completion.
Definition 4.18. Let A be a partial abelian`monoid and M an n-manifold.
We denote by C0(M; A) the quotient of kC0k(M) xk Ak under the following
relation ~: if (m1; : :;:mk) 2 C0k(M), a1; : :;:ak 2 A, m1 = m2 and a1 + a2 is
defined, then
(m1; : :;:mk; a1; : :;:ak) ~ (m2; : :;:mk; a1 + a2; : :;:ak):
Lemma 4.19. If A is a partial abelian monoid, and M is an n-dimensional
open manifold, then the inclusion C0(M; A) ,! C(M; A) is a weak equivalence.
Proof. The proof makes use of the fact that a copy of the manifold with corners
Ck(M) lies inside its interior C0k(M), so the retraction r : Ck(M) ! C0k(M) is a
k-equivariant homeomorphism onto its image. We compare via this retraction
the pushout diagram for C0k(M; A)
C0k(M) xk Tk(X; x0) [ (C0(M) xo Comp(A))k_- C0k-1(M; A)
\ \
| |
| |
| |
| |
?| ?|
C0k(M) xk Ak [ (C0(M) xo Comp(A))k ______-C0k(M; A);
and the pushout diagram for Ck(M; A)
Ck(M) xk Tk(X; x0) [ (C0(M) xo Comp(A))k__- Ck-1(M; A)
\ \
| |
| |
| |
| |
?| ?|
Ck(M) xk Ak _________________-Ck(M; A):
14
Here we denote by (C0(M) xo Comp(A))k the subspace of C(M)k xk Ak
of those labelled configurations such that several points are concentrated in
the same macroscopic location if and only if their labels are summable. The
inclusion of the space on the left hand top corner of the first diagram into th*
*at
of the second diagram is a homotopy equivalence, because r induces a common
retraction onto a copy of the second space. The same holds for the spaces on
the left hand bottom corner. We conclude by induction and the gluing lemma
[4]. __
|__|
If we regard a pointed space (A; a0) as a partial abelian monoid with x+a0 =
x as the only defined sums, for x 2 A, then C0(M; A) is the configuration space
with labels studied in [3].
Corollary 4.20. Let (A; a0) be a well-pointed space. Then for any open n-
manifold M there is a weak equivalence C0(M; A) ' C(M; A).
` __
Proof. The space A is a partial n-monoid by Comp = kFn(k)xk _ki=1A. |__|
For some background about toric varieties we refer to [7].
Corollary 4.21. If V is a projective toric variety such that H2(V ) is torsion
free, then there exists a partial discrete abelian monoid V , such that the uni*
*on
2
of some components of (V^) is homotopy equivalent to the space Rat(V ) of
based rational curves on V .
Proof. Guest has shown in [9] that if V is the fan associated to the variety
V [7] then the union of some components of C0(R2; V ) is homeomorphic to __
Rat(V ). The corollary follows from the theorem and from proposition 4.16. |__|
Remark 4.22. It is possible to define labelled configurations with support in*
* a
manifold with corners M. It is sufficient to choose an embedding M ,! M0, with
M0 open, consider the right Fn-submodule C(M) ,! C(M0) of configurations
macroscopically located at points of M, and carry through the discussion as for
open manifolds.
5 The relative case
We define relative labelled configuration spaces on relative manifolds.
Let (X; x0) be a pointed topological space. Let M be a manifold with corners
and N ,! M a cofibration such that M - N is an open manifold. We obtain
easily from 2.2 that each element c 2 C(M; X) is uniquely determined by a
finite set S(c) M, and for each P 2 S(c) a labelled tree TP as in 2.2, with the
only difference that the twigs of the tree are labelled by X - x0.
15
Definition 5.1.The based space C(M; N)(X) is the quotient C(M; X)= ~ by
the equivalence relation such that a ~ a0 if and only if S(a) \ (M - N) =
S(a0) \ (M - N) and the trees indexed by these intersections coincide. The base
point is the class [a] such that S(a) N.
If we regard pointed spaces as partial n-monoids, then the n-monoid com-
pletion induces a monad (Fn*; j*; *) on the category of pointed compactly gen-
erated spaces CG*. Each element in the completion Fn*(X) = ^Xnis represented
by a tree with vertex labels in Fn and twigs labels in X - x0. The product *
is given by grafting of trees, and the unit j* sends an element x to the trivial
tree labelled by x.
Proposition 5.2. If M is a parallelizable n-manifold, and N ,! M is a cofi-
bration such that M - N is open, then the functor C(M; N) has a structure of
right algebra over Fn*.
Proof. We need to exhibit a natural transformation : C(M; N)Fn*! C(M; N)
such that O C(M; N)j* is the identity and the diagram
C(M; N)* *
C(M; N)Fn*Fn*__________-C(M; N)Fn
| |
| |
|Fn*| ||
| |
?| ?|
C(M; N)Fn* ____________-C(M; N):
commutes. The morphism is obtained by grafting of trees. |___|
Definition 5.3.If (A; ae) is an n-monoid, and M; N are as before, then the
space C(M; N; A) of configurations in (M; N) with summable labels in A is the
coequalizer
C(M;N)ae-
C(M; N)Fn*(A) ________-C(M; N)A ........-C(M; N; A) :
A
Definition 5.4.A partial n-monoid A is good if the inclusion Comp(A) !
Fn(A) is a cofibration, and the partial composition ae : Comp(A) ! A induces a
map on the quotient Comp*(A) Fn*(A) of Comp(A).
The definition of a good framed partial n-monoid is similar. From now on
we will assume implicitly that all partial (framed) n-monoids are good.
By means of the framed Fulton-MacPherson operad we can define similarly
C(M; N; A), if M is an n-dimensional manifold with corners, N ,! M is a
cofibration, and A is a good partial framed n-monoid, and as in 4.13 we obtain:
Proposition 5.5. Define a filtration so that [a] 2 Ck(M; N; A) if and only if
k is the number of twigs of trees in S(a) \ (M - N). Then C(M; N; A) has the
weak topology with respect to the filtration and it is compactly generated.
16
Definition 5.6.If A is a partial framed n-monoid, then Bk(A) = C((Ik; @Ik)x
In-k ; A) for i = 1; : :;:n.
If A is a partial abelian monoid and (M; N) is any pair then we define the
relative labelled configuration space C0(M; N; A) as quotient of C0(M; A), by
identifying configurations that coincide on M -N. We state the relative version
of 4.19.
Proposition 5.7. If M is a manifold, N ,! M is a cofibration, and M - N is
open, then there is a weak equivalence C0(M; N; A) ' C(M; N; A).
Proof. The proof is similar to that of 4.19. In this case we use for each k
a k-equivariant_retraction rk : Ck(M) ! C0k(M) such that rk preserves
b-1(Mk - Nk ), where b : Ck(M) ! Mk is the blowdown. |___|
Corollary 5.8. If V is a projective toric variety such that H2(V ) is torsion f*
*ree,
with torus T and fan V , then there is a weak equivalence B2(V ) ' V xT ET .
Proof. Guest has shown in [9] that V xT ET is homotopy equivalent to __
C0(I2; @I2; V ). |__|
The relative version of 4.20 is:
Corollary 5.9. For any well-pointed space X there is a weak equivalence
C0(M; N; X) ' C(M; N; X).
Corollary 5.10. Let X be a well-pointed space considered as partial n-monoid.
Then there is a weak equivalence n(X) ___'-Bn(X).
Proof. The space of open configurations C0(In; @In; X) retracts onto n(X),
considered as space of configurations of a single labelled point in (In; @In).
The retraction is achieved [5] by pushing radially the particles away onto the
boundary. But the inclusion C0(In; @In; X) ,! C(In; @In; X) = Bn(X) is_a_
weak equivalence by 5.9. |__|
By means of configuration spaces we obtain the classifying space constructed
by Stasheff.
Proposition 5.11. Let (A; a0) be a well-pointed A1 space. The quotient map
C(I; {0}; A) ! C(I; @I; A) = B1(A) is canonically homeomorphic to the uni-
versal arrow E(A) ! B(A).
Proof. It is sufficient to carry out the discussion in the non-symmetric case:
in fact Ck([0; 1]) = Sk([0; 1]) x k, where Sk([0; 1]) compactifies the space of
strictly ordered maps from {1; : :;:k} to I = [0; 1].
Let Sk(I){0; 1} Sk(I) be the closure of the subspace of maps ff : {1; : :;:*
*k} !
I such that ff(1) = 0, ff(k) = 1. Its elements are described by appropriate tre*
*es
as in 2.2. For k 2, we have homeomorphisms r : Sk(I){0; 1} o Sk(0) : j,
where Sk(0) is the space of configurations in R macroscopically concentrated at
the point 0.
17
If ffi! ff 2 Sk(I){0; 1}, ffi2 C0k(I), then r(ff) = limi__ffi-ffi(0)_i(ffi(1*
*)-ffi(0)).
If fii! fi 2 Sk(0), fii2 C0k(R), then j(fi) = limi_fii-fii(0)_fii(1)-fii(0).
We have seen in 2.2 that Sk(0) = Kk is the associahedron. Under the
identification`Kk ~= Sk(I){0; 1} the Stasheff space B(A) is defined to be the
quotient of Sk(I){0; 1} x Ak-2, seen as space of forests labelled by A, under
the following steps:
1.We replace a tree on i twigs by a point having as label the action of the
tree on its twigs via Kix Ai! A.
2.We can cut twigs labelled by a0.
3.We identify any two labelled forests coinciding outside 0 and 1.
But this quotient`is exactly B1(A) = C(I; @I; A). In a similar way one shows
that E(A) = Sk(I){0; 1} x Ak-1= ~ is homeomorphic to C(I; {0}; A). In this __
case in 3 we identify forests coinciding outside 0. |__|
6 Approximation theorems
We say that a partial framed n-monoid A has homotopy inverse if the H-space
A^n has homotopy inverse.
Lemma 6.1. Let M be a connected compact n-manifold, M0 M a compact
n-submanifold, N M a closed submanifold, and A a partial framed n-monoid.
Suppose that either A has a homotopy inverse or the pair (M0; N \ M0) is
connected. Then there is a quasifibration
C(M0; N \ M0; A) ____- C(M; N; A) ____ss-C(M; M0[ N; A).
This holds in particular if A is path connected.
Proof. We follow the proof of proposition 3.1 in [5]. The space C(M; M0[N; A)
has a filtration by Ck := Ck(M; M0 [ N; A). There is a homeomorphism ffk :
ss-1(Ck - Ck-1) ~= C(M0; N \ M0; A) x (Ck - Ck-1) such that ssff-1kis the
projection onto the factor Ck - Ck-1. Choose a collared neighbourhood U of
M0 in M and a smooth isotopy retraction r : U ! M0 such that r(U \ N) N.
For each k there is an open neighbourhood Uk of Ck in Ck+1 such that r induces
a smooth isotopy retraction rk : Uk x I ! Ck, and a smooth isotopy retraction
r"k: ss-1(Uk) x I ! ss-1(Ck) covering rk. For any point P 2 Uk we need to
show that the restriction t : ss-1(P ) ! ss-1(r1(P )) of "r1is a weak homotopy
equivalence. If we identify domain and range of t to C(M0; N \ M0; A) by ffk,
then t pushes the labelled particles away from N, and adds a finite set of trees
T in proximity to N. But if the pair (M0; N \ M0) is connected, then the trees
in T can be moved continuously to N, where they vanish, and t is homotopic
to a homeomorphism. On the other hand, if A has a homotopy inverse, then t
has a homotopy inverse that pushes the particles away from N and adds some__
homotopy inverses of the trees in T in proximity to N. |__|
18
Proposition 6.2. Let A be a partial framed n-monoid. Then for i = 1; : :;:n
there are maps si : Bi-1(A) ____- Bi(A), such that si is a weak homotopy
equivalence for i > 1, and s1 is a weak homotopy equivalence if A has a homotopy
inverse.
Proof. Note that B0(A) is homotopic to the framed n-monoid completion of A.
For each i the base point of Bi(A) is the empty configuration. The translation
o1(t) : In ! RxIn-1 of the first coordinate by t induces a map sso1(t): B0(A) !
B1(A), composite of the induced map C(In; A) ________C(oi(t);A)-C(R x In-1; A) *
*and
the projection C(R x In-1; A) ! C((I; @I) x In-1; A). Then the `scanning'
map s1 is defined for x 2 B0(A) = C(In; A) by s1(x)(t) = sso1(2t-1)(x) 2
B1(A). For i > 0 the translation of the (i + 1)-th coordinate by t induces
similarly a map ssoi+1(t): Bi(A) ! Bi+1(A), and si+1 : Bi(A) ! Bi+1(A)
is given by si+1(x)(t) = ssoi+1(2t-1)(x). We define M = Ik x [0; 2] x In-k-1 ,
N = (@Ik x [0; 2] x In-k-1 ) [ (Ik x 0 x In-k-1 ), and we identify Bk(A) to
C(Ik x [1; 2] x In-k-1 ; @Ik x [1; 2] x In-k-1 ; A) via ok+1(1). We consider for
1 k n - 1 a commutative diagram
Bk(A) ____-C(M; N; A) ___-Bk+1(A)
| | ww
| | ww
sk+1|| s || www
| | ww
?| ?|
Bk+1(A) ___-P Bk+1(A) ! Bk+1(A) :
The top row is a quasifibration and the bottom row a fibration. The scanning
map s is defined on the total space C(M; N; A) by s(x)(t) = ssok+1(2t)(x) and
is consistent with sk+1. Now the space C(M; N; A) is contractible. In fact
by excision C(M; N; A) ~=C(M0; N0; A), with M0 = Rk x (-1; 2] x Rn-k-1
and N0 = M0- (M - N). Moreover there is a smooth isotopy Ht : (M; N) !
(M0; N0), such that H0 is the inclusion and H1(M) N0. For example define Ht
as the dilatation by 3t centered in (1_2; : :;:1_2; 3; 1_2; : :;:1_2), with 3 a*
*t the (k + 1)-st
position. We conclude by comparing the long exact sequences in homotopy_and
by induction on k. |__|
The spaces B0(A) = C(In; A) and nBn(A) are both fDn-algebras. The
map s : B0(A) ! nBn(A) constructed by looping and composing the scanning
maps in proposition 6.2 can be extended to a homotopy fDn-morphism by
rescaling suitably the scanning maps on the labels of trees in UfDn (B0(A)). By
6.2 we obtain:
Theorem 6.3. If A is a partial framed n-monoid, then s : B0(A) ! nBn(A)
is the group completion. If A has homotopy inverse, then s is a weak homotopy
equivalence.
Actually s is the group completion in the homotopy category of fDn-algebras.
19
Corollary 6.4. [15] If X is a well-pointed space, then s : C0(Rn; X) ! nnX
is the group completion. If X is path connected, then s is a weak homotopy
equivalence.
Proof. Consider X as a partial n-monoid as in corollary 4.20. Now B0(A) =
C(In; A) ' A^n by the same argument of proposition 4.16 . Moreover A^n '
C(Rn; A) by 4.16, C(Rn; A) ' C0(Rn; A) by 4.20 and nX ' Bn(X) by 5.10._
Now we can apply the theorem. |__|
Corollary 6.5. [9] If V is a projective toric variety such that H2(V ) is torsi*
*on
free, then s : Rat(V ) ! 2(V ) is the group completion.
Proof. Apply corollaries 4.21 and 5.8, and restrict to the relevant components._
|__|
Given a manifold M, and its tangent bundle o, there is a bundle fl =
C(o; @o; A) on M with fiber Bn(A) = C(In; @In; A), consisting of relative fiber-
wise configurations in the fiberwise one-point compactification modulo the sec-
tion at infinity (^o; 1). Whether @M is empty or not we can define a map
s : C(M; @M; A) ! (M; BnA) to the space of sections of fl. Note that if M is
parallelizable then (M; BnA) = Map(M; BnA). The scanning map s is con-
structed by the exponential map: if x 2 C(M; @M; A), then s(x) sends a point
P 2 M to the restriction of x to a small disc neighbourhood of P modulo its
boundary.
Theorem 6.6. Let A be a partial framed n-monoid. Let M be a compact con-
nected n-manifold with boundary. Then the scanning map s : C(M; @M; A) !
(M; BnA) is a weak homotopy equivalence. If A has homotopy inverse and
N is a compact connected n-manifold without boundary then s : C(N; A) !
(N; BnA) is a weak homotopy equivalence.
Proof. We follow the proof of 10.4 in [10]. There is a finite handle decomposi-
tion of M with no handles of index n. If M0 is obtained from M00by attaching
a handle_H_of_index_i , then we apply lemma 6.1 and we obtain a quasifibration
C(H; @H - @H \ M00; A) ! C(M0; @M0; A) ! C(M00; @M00; A). On the other
hand we have_a_fibration_(H=(H \M00); BnA) ! (M0; BnA) ! (M00; BnA).
But C(H; @H - @H \ M00; A) ~=Bn-i(A), and (H=(H\M00); BnA) ' iBn(A).
We compare the two sequences by the scanning maps and we conclude by propo-
sition 6.2 and induction on the number of handles. In the case of N we have __
even a handle of index n and we apply the second part of proposition 6.2. |__|
Corollary 6.7. If X is a well-pointed path connected space then s : C(S1; X) !
Map(S1; X) is a weak homotopy equivalence.
Proof. We consider X as partial framed 1-monoid as in corollary 4.20. By
corollary 5.10 B1(X) ' X. We apply the second part of the theorem, and__
note that (S1; X) ' Map(S1; X) because S1 is parallelizable. |__|
20
This answers a question raised by Stasheff in [22] p. 10. The analogous
result for C0(S1; X) is in [3].
Any partial framed n-monoid gives an approximation theorem for mapping
spaces, and the homotopy theorist is tempted to discover new examples. It might
be worth considering colimits of abelian monoids in the category of n-monoids.
Mathematisches Institut
Universit"at Bonn
Beringstrasse 1
53115 Bonn
Germany
e-mail: salvator@math.uni-bonn.de
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