C AHIERS DE
TOPOLOGIE ET GÉOMÉTRIE DIFFÉRENTIELLE CATÉGORIQUES
T EIMURAZ P IRASHVILI
On the PROP corresponding to bialgebras
Cahiers de topologie et géométrie différentielle catégoriques, tome 43, n
o3 (2002), p. 221-239
<http://www.numdam.org/item?id=CTGDC_2002__43_3_221_0>
© Andrée C. Ehresmann et les auteurs, 2002, tous droits réservés.
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Article numérisé dans le cadre du programme
ON THE PROP CORRESPONDING TO BIALGEBRAS by Teimuraz
PIRASHVILICAHIERS DE TOPOLOGIE ET
GEOMETRIEDIFFERENTIELLE CATEGORIQUES
Volume XLIII 3 (2002)
RÉSUMÉ. Un PROP A est une
cat6gorie
monoidalesym6trique
stricte avecla
propriete
suivante(cf [13]):
lesobjets
de A sont les nombres naturels etl’op6ration
monoidale est I’addition sur lesobjets.
Unealgebre
sur A est unfoncteur monoidal strict de A vers la
cat6gorie
tensorielle Vect des espaces vectoriels sur un corps commutatif k. On construit le PROPQT(as)
et onmontre que les
algebres
surQ7(as)
sont exactement lesbig6bres.
1 Introduction
A PROP is a
permutative category (A, C3),
whose set ofobjects
is the setof natural numbers and on
objects
the monoidal structure isgiven by
theaddition. An
A-algebra
is asymmetric
strict monoidal functors to the tensorcategory
of vector spaces.It is well-known that there exists a PROP whose
category
ofalgebras
isequivalent
to thecategory
ofbialgebras (=
associative and coassociative bial-gebras).
In[14]
there is adescription
of this PROP in terms ofgenerators
andrelations. Here we
give
a moreexplicit
construction of the sameobject.
Ourconstruction uses the
Quillen’s Q-construction
for doublecategories given
in[7].
The paper is
organized
as follows: In Section 2 we recall the definition of PROP and show how to obtain commutativealgebras
asF-algebras.
Here Tis the PROP of finite sets. In the next section
following
to[6]
we construct thePROP of noncommutative sets denoted
by 7(as)
and we show that7(as)- algebras
areexactely
associativealgebras.
The material of the Sections 2 and 3 are well known toexperts.
In Section 4 wegeneralize
the notion ofMackey
functor for double
categories
and in Section 5 we describe our heroQF (as),
which is the
PROP,
with theproperty
thatQF (as)-algebras
areexactely bialgebras. By
definition of PROP thecategory QF (as)
encodes the natural transformations H~n ~H~m
and relations between them. Here H runs over allbialgebras.
As asample
wegive
thefollowing application.
For anybialgebra H,
any natural number n E N and anypermutation a
En,
welet
be the
composition 03BCno03C3 * o 0394n : H ~ H,
where 0394n: H -H~n
is the(n -1 )-
th iteration of the
comultiplication ~
H (9H,
03C3* :H~n~ Hon~n
is inducedby
thepermutation
03C3, that isand
Mn : Hon
- H is the(n - l)-th
iteration of themultiplication
jj : H 0 H ~ H. Moreover let 03A6 :Gn
xGm -> Gnm
be the map constructed inProposition
5.3. Then it is a consequence of our discussion in Section5,
that for any
permutations a
EGn
and T EGm
one has theequality
Let us note that if o is the
identity,
thenT (,,id)
isnothing
but the Adamsoperation [11]
and hence our formulagives
the rule for thecomposition
ofAdams
operations.
2 Preliminaries
onPROP’s
Recall that a
symmetric
monoidalcategory
is acategory
S with a unit 0 E S and a bifunctortogether
with naturalisomorphisms
satisfying
some coherent conditions(see [8]).
If in addition aX,Y,Z,lX,
rX areidentity morphism then,
S is called apermutative category.
If S andS1
aresymmetric
monoidalcategories,
then a functor M : S -->81
is asymmetric
monoidal
functor
if there existisomorphisms
satisfying
the usualassociativity
and unit coherence conditions(see
for ex-ample [8]).
Asymmetric
monoidal functor is called strict if u X,Y isidentity
for all
X,
Y E S.According
to[13]
a PROP is apermutative category (A, []),
with the
following property:
A has a set ofobjects equal
to the set of naturalnumbers and en
objects
the bifunctor [] isgiven by
mDn = m + n. An A-algebra
is asymmetric
strict monoidal functor from A to the tensorcategory
Vect of vector spaces over a field k.
Examples. 1)
Let F be thecategory
of finite sets. Forany n > 0,
we let nbe the set
{1, ..., n}.
Hence 0 is theempty
set. We assume that theobjects
of 7 are the sets
n, n >
0. Thedisjoint
union makes thecategory 7
aPROP. It is well-known that the
category of algebras
over F isequivalent
tothe
category of
commutative and associativealgebras
with unit.Indeed,
if Ais a commutative
algebra,
then the functor£*(A) :
F --> Vect is aF-algebra.
Here the functor
£* (A)
isgiven by
For any map
f :
n ---> m, the action off
on£*(A)
isgiven by
where
Conversely,
assume T is a7-algebra.
We let A be the value of T on 1. Theunique
map 2 -7 1yields
ahomomorphism
On the other hand the
unique
map 0 --7 1yields
ahomomorphism
q : k =T (0) ---> T(1)
= A. Thepair (03BC, n)
defines on A a structure of commutative and associativealgebra
with unit. One can use the fact that T is asymmetric
strict monoidal functor to prove that T=
£*(A).
2)
Let us note that theopposite
of a PROP is still a PROP with the sameD. Hence the
disjoint
unionyields
also a structure of PROP on TOP. Thecategory of Fop-algebras
isequivalent
to thecategory of
cocommutative and coassociativecoalgebras
with counit. For any suchcoalgebra
C we let£*(C) :
Fop --> Vect be the
corresponding Fop-algebra.
Onobjects
we still have£*(C)(n)
=Cxn.
3)
We let Q be thesubcategory
of7,
which has the sameobjects
asF,
butmorphisms
aresurjections. Clearly
n is a subPROP of F andQ-algebras
are(nonunital)
commutativealgebras.
4)
We let Mon be thecategory
offinitely generated
freemonoids,
which is aPROP with
respect
tocoproduct. Similarly
thecategory
Abmon offinitely generated
free abelianmonoids,
thecategory
Ab offinitely generated
freeabelian groups and the
category
Gr offinitely generated
free groups are PROP’s withrespect
tocoproducts.
For thecategory
ofalgebras
over thesePROP’s see Theorem 5.2 and Remark 1 at the end of the paper.
5) Any algebraic theory
in the sense of Lawvere[2] gives
rise to a PROP.This
generalizes
theexamples
from1)
and4).
In the next section we
give
a noncommutativegeneralization
ofExamples 1)-3) .
3 Preliminaries
onnoncommutative sets
In this section
following
to[6]
we introduce the PROPF(as)
withproperty
that7(as)-algebras
are associativealgebras
with unit. As acategory 7(as)
is described in
[6], p.191
under the name"symmetric category" .
It is alsoisomorphic
to thecategory
AS considered in[10], [7]. Objects of F(as)
arefinite sets. So
Ob(7) = Ob(7(as)).
Amorphism
from n to m is a mapf : n -
mtogether
with a totalordering
onf -1(j)
forall j
E m.By
abuse of notation we will denote
morphisms
inF(as) by f , g
etc. Moreoversometimes we write
f for
theunderlying
map off
E7(as) .
We will alsosay that
f
is a noncommutativelifting
of a mapI f| .
In order to definethe
composition
inF(as)
we recall the definition of ordered union of ordered sets. Assume A is atotally
ordered set and for each A E A atotally
orderedset
XÀ
isgiven.
Then X =11 Xa
is thedisjoint
union of the setsXx
whichis ordered as follows. If x E
Xx and y
EXp, then x y
in X iff A 03BC ork = 03BC and x y in Xk.
If
f
EHomF(as)(n, m) and g
EHomF(as)(m, k),
then thecomposite g f
is| g || 1 as
a map, while the totalordering
in(g f)-1(i), i
E k isgiven by
theidentification
Clearly
one has theforgetful
functorF(as)
--> F . Amorphism f
in7(as)
is called a
surjection
if the mapf its
asurjection.
Anelementary surjection
is a
surjection f :
n --> m for which n - m 1.Since any
injective
map has theunique
noncommutativelifting,
we seethat the
disjoint union,
which defines thesymmetric
monoidalcategory
struc-ture in 7 has the
unique lifting
in7(as) .
Hence7(as)
is a PROP.We claim that the
category F(as)-algebras
isequivalent
to thecategory of
associativealgebras
with unit. Theonly point
here is thefollowing.
Letus denote
by TTiEI xi
theproduct
of theelements x2
E A where I is a finitetotally
ordered set and theordering
in theproduct
follows to theordering
I. Here A is an associative
algebra.
Now we have a7(as)-algebra X. (A) : 7(as) -
Vect. Here the functorX*(A)
isgiven by
the same rule as.c*(A)
in the
previous section,
but to takeTT in
the definition ofbj .
Forexample,
if
f :
4 - 3 isgiven by f ()
=f (2) = f (4)= 3, f(3) =
1 and the totalordering
inf-1 (3)
is 2 4 1then f * : A x4
-Ax3
isnothing
buta1 x a2 x a3 x a4 --> a3 x 1 x a2a4a1.
We let
Q (as)
be thesubcategory of F(as),
which has the sameobjects
asF(as),
butmorphisms
aresurjections. Clearly S2(as)
is a subPROPof F(as)
and
Q(as)-algebras
are(nonunital)
associativealgebras.
Quite similarly,
for any coassociativecoalgebra
C with counit one has aF(as)op-algebra X*(C): 7(as)°P -
Vect withX*(C)(n)
=Con
and the cate-gory
F(as)op-algebras is equivalent
to thecategory of
coassociativecoalgebras
with counit.
In order to
put bialgebras
in thepicture
we need thelanguage
ofMackey
functors.
4 On double categories and Mackey functors
Let us recall that a double
category
consists ofobjects,
a set of horizontalmorphisms,
a set of verticalmorphisms
and a set ofbimorphisms satisfying
natural conditions
[4] (see
also[7]).
If D is a doublecategory,
we letDh (resp.
DI)
be thecategory
ofobjects
and horizontal(resp. vertical) morphisms
ofD.
A Janus
functor
M from a doublecategory
D to Vect is thefollowing
data
i)
a covariant functor M* :Dh -->
Vectii)
a contravariant functorM* : (Dv)op -->
Vectsuch that for each
object
S E D one hasM*(S)
=M*(S)
=M (S).
AMackey
functor
M =(M*, M*)
from a doublecategory
D to Vect is a Janus functorM from a double
category
D to Vect such that for eachbimorphism
in Dthe
following equality
holds:Examples 1)
Let C be acategory
withpullbacks.
Then one has a doublecategory
whoseobjects
are the same as C. Moreover Morv -Morh - Mor(C),
whilebimorphisms
arepullback diagrams
in C. In this case the notion ofMackey
functorscorresponds
topre-Mackey
functors from[5]. By
abuse of notation we will still denote this double
category by
C. In what follows 7 isequipped
with this doublecategory
structure.2)
Now we consider a doublecategory,
whoseobjects
are still finitesets,
but M orv =Morh
=M or(F(as)), where F(as)
was introduced in Section 3.By
definition a
bimorphism
is adiagram
in7(as)
such that the
following
holds:i)
theimage| a of
a in 7 is apullback diagram
of sets,ii)
for all x E T the induced mapf* : 0-1 1(x)
-->0-1 (fx)
is anisomorphism
of ordered sets
iii)
for ally E S
the induced map0. f1 -1 (y)
-4f-1 (01y)
is anisomorphism
of ordered sets.
Let us note that for a bimorhism a in
F(as)
ingeneral f
o01 # 0
o11.
By
abuse of notation we will denote this doublecategory by 7(as).
It isdifferent from a double
category
considered in[7],
which is also associated to thecategory -T(as).
One observes that for any arrows
f : T -> V , 0 : S -> V in F(as)
thereexists a
bimorphism
a which hasf and 0
asedges
and it isunique
up tonatural
isomorphism. Indeed,
as a set we take U to be thepullback
and thenwe lift set maps
f 1
and01
to the noncommutative worldaccording
to theproperties ii)
andiii) . Clearly
suchlifting
exists and it isunique.
3)
We can also consider the doublecategory F(as)1
whoseobjects
are stillfinite
sets,
vertical arrows are set maps, while horizontal ones aremorphisms
from
F(as) .
Thebimorphisms
arediagrams
similar to thediagrams
in Ex-ample 2)
but suchthat 0
and01
are set maps, whilef
andfi
aremorphisms from F(as) .
Furthermore the conditionsi)
andiii)
from theprevious example
hold. We need also a double
category F(as)2
which is definedsimilarly,
butnow vertical arrows are
morphisms
fromF(as)
and horizontal ones are set maps.We have the
following diagram
of doublecategories,
where arrows areforgetful
functorsLet D be one of the double
categories
considered in(4.0) .
Abimorphism
a iscalled
elementary
if bothf and 0
areelementary surjections.
Thefollowing
Lemma for D = F was
proved
in[1].
Theproof
in other cases isquite
similarand hence we omit it.
Lemma 4.1 Let D be one
of
the doublecategories
considered in(4.0) .
Thena Janus
functor
M is aMackey functor iff
thefollowing
two conditions holdi) for
anyinjection
g : A -> B one hasM*(g)M*(g)
=idA ii) for
anyelementary birrtorphism
a one hasTheorem 4.2 Let V be a vector space, which is
equipped sirriultaneousl y
withthe structure
of
associativealgebra
with unit and coassociativecoalgebra
withcounit. Then V is a
bialgebra iff
is a
Mackey functor.
Proof.
One observes that the condition1)
of theprevious
lemmaalways
holds. On the other hand the
diagram
is a
bimorphism.
Heref -1 (1)= 11 2}, p-1 (1) = 11 2}, p- 1 (2) = {3 4}, q-1 (1)= 11 3}
andq-1(2) = {2 4}. Clearly f* : Vx2
- V is themultiplication p
on V andf * :
V -V x2
is thecomultiplication A
onV,
while p. =
(IL
0it)
o T2,3 andq*
= T2,3 0 A xA,
where T2,3 :Vx4
->Vx4
permutes the second and the third coordinates. Hence V is abialgebra
iffthe condition
ii)
of theprevious
lemma holds for cx. Since bothX* (V)
andX* (V)
senddisjoint
union to tensorproduct
the result follows from Lemma 4.1.Addendum. For a cocommutative
bialgebra
C theMackey
functorX(C)
factors
through
the doublecategory F(as)1,
for a commutativebialgebra
Athe
Mackey
functorX (A)
factorsthrough F(as)2
and in the case of commu-tative and cocommutative
bialgebra
H one has theMackey
functor£(H) :
7 -> Vect.
5 The construction of QF(as)
Let D be one of the double
categories
considered inExamples 1 )-3) . Clearly categories
D’ andDh
have the same class ofisomorphisms,
which we callisomorphisms of D.
We letQD
be thecategory
whoseobjects
are finitesets,
while themorphisms
from T to S areequivalence
classes ofdiagrams:
Here
f
EDh
is a horizontalmorphism and 0
E DV is a verticalmorphism.
For
simplicity
such data will be denotedby T 40U fl
S. Twodiagrams
0 f 01 1f
T--
U->
S andT -- Ul A
S’ areequivalent
if there exists a commutativediagram
such that h is an
isomorphism.
Thecomposition
ofT t,
Uf->
S andS -- Y
Vg--> R
inQD
isby
definitionT --Y0 W -->gf1 R,
whereis a
bimorphism
in D. Oneeasily
checks thatQD
is acategory
and for anyobject
S thediagram S --ls
S-->ls
S is anidentity morphism
inQD.
Clearly
thedisjoint
unionyields
a structure of PROP onQD
and 0 is notonly
a unitobject
withrespect
to this monoidal structure, but also a zeroobject.
For a horizontal
morphism f :
S - T in D we leti* (f) :
S’ -3 T be thefollowing morphism
inQD:
Similarly,
for a verticalmorphism 0 :
S e T we leti* (f) :
T --> S be thefollowing morphism
inQD:
In this way one obtains the
morphisms
of PROP’S: i,, : D -->QD
and i* : Dop -->QD.
Remark. The construction of
QD
is aparticular
case of thegeneralized Quillen Q-constuction [16]
consideredby
Fiedorowicz andLoday
in[7].
Thefollowing
lemma is a variant of a result of[9].
Lemma 5.1 The
category of Mackey functors from
D to Vect isequivalent
to the
category of functors
M :QD -->
Vect.Proof.
Let M :QD -
Vect be a functor. For any arrowf :
S --> Twe
put M*(f)
:=M(i*(f))
andM*(f)
:=M(i*(f)).
In this way weget
aMackey
functor on D.Conversely,
if M is aMackey
functor onD,
then weput
One
easily
shows that in this way weget
a covariant functorQD
to Vect and theproof
is finished.By applying
theQ-construction
to thediagram (4.0)
one obtains thefollowing (noncommutative) diagram
of PROP’s:The
following
theoremgives
the identification of the terms involved in thediagram, except
forQ(F(as)).
Theorem
5.2 i)
Thecategory of Q(F(as)) -algebras
isequivalent
to the cat-egory
of bialgebras.
ii)
Thecategory Q(F(as)1)-algebras is equivalent
to thecategory of
co-commutative
bialgebras
andQ(F(as)1)
isisomorphic
to the PROP MonDp.iii)
Thecategory of Q(F(as)2)-algebras
isequivalent
to thecategory of
commutative
bialgebras
andQ(F(as)2)
isisomorphic
to the PROP Mon.iv)
Thecategory of Q(F)-algebras
isequivalent
to thecategory of
cocom-mutative and commutative
bialgebras
andQ(.F)
isisomorphic
to the PROPAbmon.
Proo, f.
Theorem 4.2together
with Lemma 5.1 shows that anybialgebra
Vgives
rise toX(V)-algebra. Conversely
assume M is aQ(F(as))-algebra
andlet V =
M (1) .
ThenM o i*
is aF(as)-algebra
and M o i* is aF(as)op-algebra.
Thus M carries natural structures of associative
algebra
and coassociativecoalgebra.
Since M =(M o i*, M o i*)
is aMackey
functoron 7(as) ,
it followsfrom Theorem 4.2 that V is indeed a
bialgebra.
To prove theremaining parts
of thetheorem,
let us observe that(Q(F(as)2))op = Q(F(as)1),
whereequivalence
isidentity
onobjects
and sendsT --0
Uf-->
S tos--f
U0--> T .
We now show that
Q(F(as)2)=
Mon. The main observation here is the fact that iff :
X -->Si jl S2
is amorphism
inF(as)
thenf = fIll f2
inthe
category F(as),
wherefi
as a map is the restriction off
onf-l(8i),i
=1, 2.
Sincefi-1 (y) = f -1 (y)
for all y Ef-1 (Si)
we can take the same totalordering
infi-1 (y)
to turnf
Z into amorphism
inT(as).
A conclusion of this observation is the fact thatdisjoint
union defines notonly
asymmetric
monoidal
category
structure but it is thecoproduct
inQ(F(as)2)’ Clearly
n is an n-fold
coproduct
of 1. On the otherhand,
we may assume that theobjects
of Mon are naturalnumbers,
while the set ofmorphisms
fromk to n is the same as
HomMon(Fk, Fn),
whereFn
is the free monoid on ngenerators.
This set can be identified with the set ofk-tuples
of words onn variables x1, ... , xn. Since
Q(F(as)2)
and Mon arecategories
with finitecoproducts
and anyobject
in bothcategories
is acoproduct
of somecopies
of
1,
we needonly
toidentify
the set ofmorphisms originating
from 1. Amorphism
1 --> n inQ(F(as)2)
is adiagram
1 --0 U -4 n,where 0
is a map ofnoncommutative sets. We can associate to this
morphism
a word w oflength
m on n variables x1, ... , xn . Here m =
Card(U)
and the i-thplace
of w isx f(yi) ,
where U ={y1
...ym}.
In this way one seesimmediately
thatthis
correspondence
defines theequivalence
ofcategories Q(7(as)2) E£
Mon.We refer the reader to
[1]
for the fact thatQ(7)
isequivalent
to Abmon.Argument
in this case is evensimpler
than theprevious
one and can besketched as follows. Since the PROP
Q(7)
isisomorphic
to itsopposite disjoint
unionyields
notonly
thecoproduct
inQ(F)
but also theproduct.
Next, morphisms
1 --> 1in Q(7)
arediagrams
of maps 1-- U -->1,
whoseequivalence
class iscompletely
determinedby
thecardinality
of U. Thisgives
identification of
morphisms
from 1 --> 1 with natural numbers and theproof
is done.
Thus the above
diagram
of PROP’s isequivalent
to thediagram
Here Mon --> Abmon is
given by
abelization functor. Let us note thatQ(J’(as))
and Abmon are self dualPROP’S,
and the arrows aresurjection
on
morphisms.
If one looks atendomorphisms
of 1 we see that the endo-morphism
monoidEndc(1)
for C =Monop, Mon,
Abmon isisomorphic
to the
multiplicative
monoid of natural numbers. Thiscorresponds
to thefact that the
operations w(n,u)
from the introduction for commutative or cocommutativebialgebras
areindependent
of o and BI1n o BI1m = wnm[11].
The
following proposition
describes theendomorphism
monoidEndc (I)
for C =
Q(7(as)).
Let n E N be a natural number and let a E
Qn
be apermutation.
HereQn
is the group ofpermutations
on n letters. We let[o]
be themorphism
n --> 1 in
F(as) corresponding
to theordering o(1) o(2)
...o(n).
For
example [ion],
orsimply [id]
denotes themorphism n
e 1 inF(as) corresponding
to theordering
1 2 ... n. Moreover we let(n, a) :
1 e 1be the
morphism in Q(F(as)) corresponding
to thediagram 1 --[o]
n[id]-->
1.Proposition
5.3 The monoidof endomorphisms of 1
EQ(F(as))
is iso-rriorphic
to the monoidof pairs (n, o),
where a E6n
and n EN,
with thefollowing multiplication
Here
is a rriap, zuhich is
defined by
Proof.
Amorphism
1-t 1 inQ(F(as))
is adiagram 1t,
Uf--> 1,
where0
andf are morphisms
of noncommutative sets. Hence U has two totalorderings corresponding to 0
andf .
We willidentify
U to n, viaordering corresponding
tof .
Here n is thecardinality
of U. We denote the first(resp.
the
second, ...)
element in theordering corresponding to 0 by o(1) (resp.
o(2), ... ).
In this way weget
apermutation
EQn.
Thus anymorphism
1--> 1 in
Q(F(as))
is of the form(n, o).
In order toidentify
thecomposition
law it is
enough
to note thefollowing
two facts:i)
Thediagram
is a
bimorphism
inQ(F(as)).
Heref and g
aregiven by
for i E m
and j
E n.ii)
One has[@(o, T)] = [T]
o g and[idn] o f
=(id j.
We now
give
an alternativedescription
of the function 4(D. Letbe a map
given by
where
o(m1, ..., mn) permutes
the n blocksaccording
to p.Moreover,
forany
integers
n and m we letbe the
bijection corresponding
to thefollowing ordering
of the Cartesianproduct:
Similarly,
we letbe the
bijection corresponding
to thefollowing ordering
of the Cartesianproduct:
Then we
put @(n, m)
:=I-1
o II E6nm.
It is not too difficult to see that@(n, m)= @(In, Im)
andRemarks.
1)
It is well known that the PROPcorresponding
to cocommu-tative
Hopf algebras
is Gr°p(see
nextremark),
the PROPcorresponding
tocommutative
Hopf algebras
isGr,
while the PROPcorresponding
to com-mutative and cocommutative
Hopf algebras
is Ab. Of course thecategory
ofHopf algebras
are alsoalgebras
over somePROP,
which can beeasily
described via
generators
and relations[14].
Anexplicit description
of thisparticular
PROP will be thesubject
of aforthcoming
paper.2)
Let A be a cocommutativeHopf algebra.
Since 0 is aproduct
in thecategory Coalg
of cocommutativecoalgebras,
A is a groupobject
in thiscategory.
On the other hand any groupobject
in anycategory
A with finiteproducts gives
rise to the model in A of thealgebraic theory
of groups in thesense of Lawvere
[2].
But thealgebraic theory
of groups isnothing
but Gropand hence we have the functor
X(A) :
Grop--> Coalg,
whichassigns
Axn ton > . Here n > is a free group on x 1, Xn. Moreover it
assigns
J..L to themorphism
1 > --> 2 >given by
xl H xlx2.Similarly x(A) assigns
A to
F(P)
thehomomorphism
2 > --> 1 >given by
x1, x2 H Xl’ Ofcourse it
assigns
theantipode
S : A --> A to xl Hx1-1. Having
these facts in mind oneeasily
describes the action ofX(A)
on morecomplicate morphisms.
For
example
one check’s thatX(A) assigns
to the
morphism
2 > --> 3 >corresponding
to thepair
of words(x1-1 x2x1, zfz3).
Here r2,3permutes
the second and third coordinates. Con-versely
any linear mapAxn
eAxm
constructedusing
the structural data of a cocommutativeHopf algebra
A iscomming
in this way. Hence to check whether acomplicated diagram involving
such maps commutes it isenough
tolook to the
corresponding diagram
inGr,
which isusually simpler
to handle.3)
It is well known that themorphism
n --> m in Abmon can be identified with(m
xn)-matrices
over natural numbers. Under this identification theequivalence Q(F) =
A bmon isgiven by assigning
the matrix whose(i, j )-
component is the
cardinality of f-1 (j) ^ g-1 (i),
1 i m,1 j
n to thediagram 11 f-, X -.!4
m. It is less known that themorphisms n
--> m in Moncan be described via shuffles. In order to
explain
this connection let us start withparticular
case. Consider a wordx2yxy3x2 of bidegree (5,4).
It defines amorphism
1--> 2 in Mon. One associates a(5, 4)-shuffle (1, 2, 4, 8, 9, 3, 5, 6, 7)
to this
word,
whose first five values arejust
the numbers ofplaces
where xlies.
Similarly morphisms
n --> rrt in Mon are in1-1-correspondence
withcollections
{A
=(aij), (cp1,"’, &n)},
where A is an(m
xn)-matrix
overnatural numbers and cpi is a
(ail, - - - , aim)-shuffle, i
=1, ... ,
n. The functor Mon - Abmoncorresponds
toforgetting
the shufHes. Now combine this observation withProposition
5.3 toget
thedescription
ofmorphisms
n --> min
6(JF(as))
as collections{A
=(aij), (&1, ...&n)},
where a =(aij, Uij)
andaij is a natural
number,
while oij E6aii
is apermutation
andfinally
cpi is a(ail, - - - aim)-shuffle.
4) Recently
Sarah Whitehouse([17], [3])
defined the action ofGk+1
onAxk
for any commutative or cocommutative
Hopf algebra
A.Actually
sheimplic- itly
constructed the grouphomomorphism
where
Gk
is theautomorphism
group of k >. Then the action of x EGk+1
on
Axk
is obtainedby applying
the functorX(A)
to§k(x) .
The homomor-phism §k
isgiven by
for I and
Here
EGk+1
is thetransposition (i, i+1),
1 i k. Thehomomorphisms
§k, k>
0 are restrictions of afunctor § :
J’ -->Gr,
which isgiven
as follows.For a set X the
group § (X)
isgenerated by symbols
x, y >, x, y E X modulo the realtions6 Generalization for operads
Let P be an
operad
of sets[12].
Let us recall that then P is a collection ofGn-sets P(n), n>
0together
with thecomposition
lawand an element e E
P (1) satisfying
someassociativity
and unite conditions[12].
We will assume thatP(O) =
*.Any
set Xgives
rise to anoperad EX,
for whichEX (n)
=Maps (Xn, X) .
AP-algebra
is a set Xtogether
with a
morphism
ofoperads P
eEX.
We letP-Alg
be thecategory
of P-algebras.
Theforgetfull
functorP-Alg
--> Sets has the leftadjoint
functorFP :
Sets eP-Alg
which isgiven by
We let
Free(P)
be the fullsubcategory
ofP-Alg
whoseobjects
areFP(n),
n>0.
Now we introduce the
category F(P).
For any mapf :
n --> rrL oneputs
Here
|S denotes
thecardinality
of a set S. Thecategory F(P)
has the sameobjects
asY,
while themorphisms
from n --> m inF(P)
arepairs ( f , w f ),
where
f :
n -->n->m is a map
m is a map andwf
=(w1f , ... , wmf)
EPf -
f If(f , wf)
and(g, wg) : m - k
aremorphisms
in7(P)
then thecomposition (g, wg)
o( f, wf )
is a
pair (h, wh),
where h =g f
and for each 1 i k one hasHere g-1 (i)
={j1, ... , js}.
This construction goes back toMay
and Thoma-son
[15].
One observes that if P = as, then
7(P)
isnothing
butF(as),
whileFree(P)
isequivalent
to thecategory
offinitely generated
free monoids. Hereas is the
operad given by as(n) = Gn
for all n > 0and q
is the same asin
(5.4).
Thusas-algebras
are associative monoids. We now show how togeneralize
Theorem 5.2ii)
forarbitrary operads.
Let
7(P)2
be the doublecategory,
whoseobjects
aresets,
horisontalarrows are set maps and vertical arrows are
morphisms
from7(P).
Doublemorphisms
arepulback diagrams
of setstogether
withlifting of g
andf
inF(P).
Hence the elementswf
EP f
andw9 E
Pg
aregiven.
Onerequires
that these elements arecompatible
We claim that the
category Q(F(P)2)
andFree(P)
areequivalent.
Onobjects
one
assignes FP (n)
to n. Bothcategories
in thequestion
posses finitecoprod-
ucts and thus one needs
only
toidentify morphisms
from 1. Let1 -- w
mf-->
Xbe a
morphism in Q(F(P)2). By
definition w EP(m)
andf
E Xn. Thusit
gives
an element inFp(X)
and therefore amorphism Fp(1)
-->Fp(X)
in
Free (X) .
It is clear that in this way one obtainsexpected equivalence
ofcategories.
Any
setoperad
Pgives
rise to the linearoperad k[P],
which isspanned
on P.
Clearly
thedisjoint
unionyields
a structure of PROP onF(P)
andF(P)-algebras
arenothing
butk[P]-algebras
in the tensorcategory
Vect.We leave as an exercise to the interested readers to show that the
Q-
construction of Section 5 and the notion of the
bialgebra
have the canonicalgeneralizations
for anyoperad
P.Acknowledgments
This work was written
during
my visit at theSonderforschungsbereich
derUniversitat Bielefeld. I would like to thank Friedhelm Waldhausen for the invitation to Bielefeld. It is a
pleasure
toacknowledge
varioushelpful
dis-cussions I had with V.
Franjou
and J.-L.Loday
on thesubject
and also for invitations in Nantes and LePouliguen,
where this work was started. Thanks to Ross Street on his comments. The author waspartially supported by
thegrant
INTAS-99-00817 andby
the TMR networkK-theory
andalgebraic
groups, ERB FMRX CT-97-0107.
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email: