Some remarks on connected coalgebras

11 Pages·2007·0.243 MB·English

by  A. Ardizzoni, Claudia Menini

SOME REMARKS ON CONNECTED COALGEBRAS
A.ARDIZZONIANDC.MENINI
Dedicated to Freddy Van Oystaeyen, on the occasion of his sixtieth birthday
Abstract. Inthispaperweintroducethenotionsofconnected,0-connectedandstrictlygraded
coalgebraintheframeworkofanabelianmonoidalcategoryMandweinvestigatetherelations
betweentheseconcepts. Werecoverseveralresults,involvingthesenotions,whicharewellknown
inthecasewhenMisthecategoryofvectorspacesoverafieldK. Inparticularwecharacterize
whena0-connectedgradedbialgebraisabialgebraoftypeone.
Introduction
Let M be a coabelian monoidal category such that the tensor product commutes with direct
sums. Givenagradedcoalgebra(C =⊕ C ,∆,ε)inM,wecanwrite∆ asthesumofunique
n∈N n |Cn
components ∆ : C → C ⊗C where i+j = n. The coalgebra C is defined to be a strongly
i,j i+j i j
N-graded coalgebra (see[AM1, Definition2.9])when∆C :C →C ⊗C isamonomorphismfor
i,j i+j i j
every i,j ∈N. The associated graded coalgebra
C∧ C C∧ C∧ C
gr E =C⊕ E ⊕ E E ⊕··· ,
C C C∧ C
E
for a given subcoalgebra C of a coalgebra E in M, is an example of strongly N-graded coalgebra
(see [AM2, Theorem 2.10]).
A graded coalgebra (C = ⊕ C ,∆ ,ε ) in a cocomplete monoidal category M is called
n∈N n C C
0-connected whenever ε iC :C →1 is an isomorphism where iC :C →C denotes the canonical
C 0 0 0 0
injection. C is called strictly graded whenever it is both strongly N-graded and 0-connected. The
associated graded coalgebra gr C of a coaugmented coalgebra C in M is an example of a strictly
1
graded coalgebra (see Theorem 2.10). We also introduce the notion of connected coalgebra in M
(see Definition 2.1).
In Theorem 2.11 we prove the following result. Let (C = ⊕ C ,∆ ,ε ) be a 0-connected
n∈N n C C
graded coalgebra in a cocomplete coabelian monoidal category M. Then
(cid:161) (cid:162)
1) (C,∆ ,ε ),u =iC is a connected coalgebra where u :=iCε−1 :1→C;
C C C 1 C 0 0
2) C ∧ C =C ⊕P (C), where P (C) denotes the primitive part of C.
0 C 0 0
Moreover, if M is also complete and satisfies AB5, the following assertions are equivalent:
(a) C is a strongly N-graded coalgebra;
(b) C =P (C).
1
Thisresultisthenappliedtothefollowingsetting. LetH beabraidedbialgebrainacocomplete
and complete abelian braided monoidal category (M,c) satisfying AB5. Assume that the tensor
product commutes with direct sums and is two-sided exact. Let M be in HMH. Let T =T (M)
H H H
betherelativetensoralgebraandletTc =Tc(M)betherelativecotensorcoalgebraasintroduced
H
in [AMS1]. In [AM1], we proved that both T and Tc have a natural structure of graded braided
bialgebraandthatthenaturalalgebramorphismfromT toTc,whichcoincideswiththecanonical
injections on H and M, is a graded bialgebra homomorphism. Thus its image is a graded braided
1991 Mathematics Subject Classification. Primary18D10;Secondary16W30.
Key words and phrases. MonoidalCategories,ConnectedCoalgebras,GradedCoalgebras,BraidedBialgebras.
This paper was written while the authors were members of G.N.S.A.G.A. with partial financial support from
M.I.U.R..
1
2 A.ARDIZZONIANDC.MENINI
bialgebra which we denote by H[M] and call, accordingly to [Ni], the braided bialgebra of type one
associated to H and M (see [AM1, Definition 6.7]).
Let now (B,m ,u ,∆ ,ε ) be a braided graded bialgebra in (M,c). Assume that B is 0-
B B B B
connected as a coalgebra. Then, by the foregoing, ((B,∆ ,ε ),u ) is a connected coalgebra. We
B B B
prove(seeTheorem2.13)thatB isthebraidedbialgebraoftypeoneB [B ]associatedto B and
0 1 0
B if and only if
1
(⊕ B )2 =⊕ B and P (B)=B .
n≥1 n n≥2 n 1
Therefore TOBAs, as introduced in [AG, Definition 3.2.3], are exactly the braided bialgebras of
type one in M=HYD which are 0-connected.
H
1. Preliminaries and Notations
Notations. Let [(X,i )] be a subobject of an object E in an abelian category M, where
X
i =iE :X (cid:44)→E is a monomorphism and [(X,i )] is the associated equivalence class. By abuse
X X X
of language, we will say that (X,i ) is a subobject of E and we will write (X,i ) = (Y,i ) to
X X Y
mean that (Y,i )∈[(X,i )]. The same convention applies to cokernels. If (X,i ) is a subobject
Y X X
of E then we will write (E/X,p )=Coker(i ), where p =pE :E →E/X.
X X X X
Let (X ,iY1) be a subobject of Y and let (X ,iY2) be a subobject of Y . Let x : X → X and
1 X1 1 2 X2 2 1 2
y :Y →Y bemorphismssuchthaty◦iY1 =iY2 ◦x. Thenthereexistsauniquemorphism,which
1 2 X1 X2
we denote by y/x= y :Y /X →Y /X , such that y ◦pY1 =pY2 ◦y:
x 1 1 2 2 x X1 X2
X (cid:31)(cid:127) iYX11 (cid:47)(cid:47)Y pYX11 (cid:47)(cid:47) Y1
1 1 X1
x y y
x
X(cid:178)(cid:178) (cid:31)(cid:127) iYX22 (cid:47)(cid:47)Y(cid:178)(cid:178) pYX22 (cid:47)(cid:47) Y(cid:178)(cid:178)2
2 2 X2
δ will denote the Kronecker symbol for every u,v ∈N.
u,v
1.1. Monoidal Categories. Recall that (see [Ka, Chap. XI]) a monoidal category is a category
M endowed with an object 1 ∈ M (called unit), a functor ⊗ : M×M → M (called tensor
product), and functorial isomorphisms a : (X ⊗Y)⊗Z → X ⊗(Y ⊗Z), l : 1⊗X → X,
X,Y,Z X
r : X ⊗1 → X, for every X,Y,Z in M. The functorial morphism a is called the associativity
X
constraint and satisfies the Pentagon Axiom, that is the following relation
(U ⊗a )◦a ◦(a ⊗X)=a ◦a
V,W,X U,V⊗W,X U,V,W U,V,W⊗X U⊗V,W,X
holds true, for every U,V,W,X in M. The morphisms l and r are called the unit constraints and
they obey the Triangle Axiom, that is (V ⊗l )◦a =r ⊗W, for every V,W in M.
W V,1,W V
A braided monoidal category (M,c) is a monoidal category (M,⊗,1) equipped with a braiding
c, that is a natural isomorphism c :X⊗Y −→Y ⊗X for every X,Y,Z in M satisfying
X,Y
c =(c ⊗Y)(X⊗c ) and c =(Y ⊗c )(c ⊗Z).
X⊗Y,Z X,Z Y,Z X,Y⊗Z X,Z X,Y
For further details on these topics, we refer to [Ka, Chapter XIII].
It is well known that the Pentagon Axiom completely solves the consistency problem arising
out of the possibility of going from ((U ⊗V)⊗W)⊗X to U ⊗(V ⊗(W ⊗X)) in two different
ways (see [Mj1, page 420]). This allows the notation X ⊗···⊗X forgetting the brackets for
1 n
any object obtained from X ,···X using ⊗. Also, as a consequence of the coherence theorem,
1 n
the constraints take care of themselves and can then be omitted in any computation involving
morphisms in a monoidal category M.
Thus, for sake of simplicity, from now on, we will omit the associativity constraints.
The notions of algebra, module over an algebra, coalgebra and comodule over a coalgebra can be
introducedinthegeneralsettingofmonoidalcategories. GivenanalgebraAinMoncandefinethe
categories M, M and M of left, right and two-sided modules over A respectively. Similarly,
A A A A
SOME REMARKS ON CONNECTED COALGEBRAS 3
given a coalgebra C in M, one can define the categories of C-comodules CM,MC,CMC. For
more details, the reader is referred to [AMS2].
Definitions 1.2. Let M be a monoidal category.
We say that M is a coabelian monoidal category if M is abelian and both the functors
X⊗(−):M→M and (−)⊗X :M→M are additive and left exact, for any X ∈M.
1.3. Let M be a coabelian monoidal category.
Let (C,iE) and (D,iE) be two subobjects of a coalgebra (E,∆,ε). Set
C D
E E
∆ :=(pE ⊗pE)∆:E → ⊗
C,D C D C D
(C∧ D,iE )=ker(∆ ), iE :C∧ D →E
E C∧ED C,D C∧ED E
E (cid:161) (cid:162) E
( ,pE )=Coker iE =Im(∆ ), pE :E →
C∧ D C∧ED C∧ED C,D C∧ED C∧ D
E E
Moreover, we have the following exact sequence:
iE pE E
(1) 0−→C∧ D C−∧→ED E C−∧→ED −→0.
E C∧ D
E
Assumenowthat(C,iE)and(D,iE)aretwosubcoalgebrasof(E,∆,ε). Since∆ ∈EME,itis
C D C,D
straightforwardtoprovethatC∧ D isacoalgebraandthatiE isacoalgebrahomomorphism.
E C∧ED
Let(C,iE)beasubobjectofacoalgebra(E,∆,ε)inacoabelianmonoidalcategoryM. Wecan
C (cid:161) (cid:162)
define (see [AMS2]) the n-th wedge product C∧En,iE of C in E where iE : C∧En → E.
C∧En C∧En
By definition, we have
C∧E0 =0 and C∧En =C∧En−1∧ C, for every n≥1.
E
One can check that C∧Ei∧E C∧Ej =C∧Ei+j for every i,j ∈N.
Assume now that (C,iE) is a subcoalgebra of the coalgebra (E,∆,ε). Then there is a (unique)
C
coalgebra homomorphism
iC∧En+1 :C∧En →C∧En+1, for every n∈N.
C∧En
such that iE ◦iC∧En+1 =iE .
C∧En+1 C∧En C∧En
1.4. Graded Objects. Let (X ) be a sequence of objects a cocomplete coabelian monoidal
n n∈N
category M and let
(cid:77)
X = X
n
n∈N
be their coproduct in M. In this case we also say that X is a graded object of M and that the
sequence (X ) defines a graduation on X. A morphism
n n∈N
(cid:77) (cid:77)
f :X = X →Y = Y
n n
n∈N n∈N
is called a graded homomorphism whenever there exists a family of morphisms (f :X →Y )
n n n n∈N
such that f =⊕ f i.e. such that
n∈N n
f ◦iX =iY ◦f , for every n∈N.
Xn Yn n
We fix the following notations. Throughout let
pX :X →X and iX :X →X
n n n n
be the canonical projection and injection respectively, for any n∈N.
Given graded objects X,Y in M we set
(X⊗Y) =⊕ (X ⊗Y ).
n a+b=n a b
ThenthisdefinesagraduationonX⊗Y whenever thetensorproductcommuteswithdirectsums.
4 A.ARDIZZONIANDC.MENINI
1.5. Let M be a coabelian monoidal category such that the tensor product commutes with direct
sums.
Recall that a graded coalgebra in M is a coalgebra (C,∆,ε) where
C =⊕ C
n∈N n
is a graded object of M such that ∆ : C → C ⊗C is a graded homomorphism i.e. there exists a
family (∆ ) of morphisms
n
n∈N
∆C =∆ :C →(C⊗C) =⊕ (C ⊗C ) such that ∆=⊕ ∆ .
n n n n a+b=n a b n∈N n
We set
(cid:195) (cid:33)
∆C =∆ := C ∆→a+b (C⊗C) ω→aC,,bC C ⊗C .
a,b a,b a+b a+b a b
A homomorphism f : (C,∆ ,ε ) → (D,∆ ,ε ) of coalgebras is a graded coalgebra homomor-
C C D D
phism if it is a graded homomorphism too.
Definition 1.6. [AM1, Definition 2.9] Let (C = ⊕ C ,∆,ε) be a graded coalgebra in M. In
n∈N n
analogy with the group graded case (see [NT]), we say that C is a strongly N-graded coalgebra
whenever
∆C :C →C ⊗C is a monomorphism for every i,j ∈N,
i,j i+j i j
where ∆C is the morphism defined in Definition 1.5.
i,j
2. Connected coalgebras
Definitions2.1. LetMbeacoabelianmonoidalcategory. Acoaugmentedcoalgebra ((C,∆,ε),u)
in M consists of a coalgebra (C,∆,ε) endowed with a coalgebra homomorphism u:1→C called
coaugmentation ofC. Notethatuisamonomorphismasεu=Id . Givenacoaugmentedcoalgebra
1
((C,∆,ε),u) define
α :=(C⊗u )◦r−1+(u ⊗C)◦l−1−∆ :C →C⊗C,
C C C C C C
(cid:161) (cid:162)
P (C),i =ker(α ).
P(C) C
(cid:161) (cid:162)
P (C),i is called the primitive part of the coaugmented coalgebra C.
P(C)
A connected coalgebra in M is a coaugmented coalgebra ((C,∆,ε),u) in M such that
l−i→m(1∧nC)n∈N =C.
Remark 2.2. Let M be the category of vector spaces over a field K and let ((C,∆,ε),u) be a
connectedcoalgebrainMaccordinglytothepreviousdefinition. ThenC :=Corad(C)⊆Im(u)
(0)
(see e.g. [AMS1, Lemma 5.2]) and hence C =Im(u) so that C is connected in the usual sense.
(0)
∧n
On the other hand, since C = lim(C C) it is clear that an ordinary connected coalgebra C is
−→ (0) n∈N
also a connected coalgebra in M.
Remark 2.3. Let C be a connected coalgebra in the monoidal category of vector spaces over a
field K. Then, the cotensor coalgebra Tc = Tc(M) is strongly N-graded and connected for every
C
C-bicomodule M. Nevertheless C needs not to be K, in general.
Question 2.4. Let M be a cocomplete coabelian monoidal category and let ((C,∆,ε),u) be a
connectedcoalgebrainM. LetM beaC-bicomoduleinM. Isittruethatthecotensorcoalgebra
Tc(M) is a connected coalgebra?
C
Lemma 2.5. Let (C,u ) be a coaugmented coalgebra and let f : C → D be a coalgebra homo-
C
morphism in a coabelian monoidal category M. Then (D,u ) is a coaugmented coalgebra where
D
u =f ◦u . Moreover
D C
α ◦f =(f ⊗f)◦α .
D C
SOME REMARKS ON CONNECTED COALGEBRAS 5
Proof. Clearly (D,u ) is a coaugmented coalgebra. Moreover, we have
D
(cid:163) (cid:164)
α ◦f = (D⊗u )◦r−1+(u ⊗D)◦l−1−∆ ◦f
D D D D D D
= (D⊗f ◦u )◦r−1◦f +(f ◦u ⊗D)◦l−1◦f −∆ ◦f
C D C D D
= (D⊗f ◦u )◦(f⊗1)◦r−1+(f ◦u ⊗D)◦(1⊗f)◦l−1−(f⊗f)◦∆
(cid:163) C C C (cid:164) C C
= (f⊗f)◦ (C⊗u )◦r−1+(u ⊗C)◦l−1−∆ =(f ⊗f)◦α .
C C C C C C
(cid:164)
Lemma 2.6. Let iE : F → E and iE : G → E be monomorphisms which are coalgebra homomor-
F G
phisms in a coabelian monoidal category M. Then
(cid:181) (cid:182)
F ∧ G F ∧ G F ∧ G
E ,Fρ : E →F ⊗ E
G F∧EG G G
G
is a left F-comodule where Fρ is uniquely defined by
F∧EG
G
(cid:181) (cid:182)
F ∧ G
Eρ = iE ⊗ E ◦Fρ .
F∧EG F G F∧EG
G G
Furthermore the following diagram
F ∧ G ∆F∧EG (cid:47)(cid:47)(F ∧ G)⊗(F ∧ G)
E E E
pF∧EG
G (cid:178)(cid:178)
F∧GEG (F∧EG)⊗pFG∧EG
FρF∧EG
G (cid:178)(cid:178) (cid:178)(cid:178)
F ⊗ F∧GEG iFF∧EG⊗F∧GEG (cid:47)(cid:47)(F ∧E G)⊗ F∧GEG
is commutative and
1ρ =l−1
1∧EG 1∧EG
G G
whenever F =1.
Proof. The first part of the statement follows by [Ar, Lemma 2.14].
Let us prove the commutativity of the diagram. We have
(cid:181) (cid:182) (cid:181) (cid:182)
F ∧ G F ∧ G
iE ⊗ E ◦ iF∧EG⊗ E ◦Fρ ◦pF∧EG
F∧EG G F G F∧GEG G
(cid:181) (cid:182)
F ∧ G
= iE ⊗ E ◦Fρ ◦pF∧EG =Eρ ◦pF∧EG
F G F∧EG G F∧EG G
G G
(cid:179) (cid:180) (cid:179) (cid:180)
= E⊗pF∧EG ◦Eρ = iE ⊗pF∧EG ◦∆ .
G F∧EG F∧EG G F∧EG
Since the tensor product is left exact, then iE ⊗ F∧EG is a monomorphism so that we obtain
F∧EG G
the commutativity of the diagram. Finally, since
(cid:181) (cid:182)
F ∧ G
l−1 = ε ⊗ E ◦Fρ and ε =Id ,
F∧EG F G F∧EG 1 1
G G
when F =1 we obtain the last equality in the statement. (cid:164)
(cid:161) (cid:162)
Lemma 2.7. Let E,u =iE be a coaugmented coalgebra in a coabelian monoidal category M.
E 1
Then
(cid:179) (cid:180)
1∧nE,u1∧nE =i11∧nE
6 A.ARDIZZONIANDC.MENINI
is a coaugmented coalgebra for every n ∈ N. Furthermore, for every n ∈ N, there exists a unique
morphism τn :1∧nE+1 →1∧nE ⊗1∧nE such that the following diagram
1∧nE+1
τn α1∧nE+1
(cid:117)(cid:117) (cid:178)(cid:178)
1∧nE ⊗1∧nE (cid:47)(cid:47)1∧nE+1 ⊗1∧nE+1
i11∧∧nEnE+1⊗i11∧∧nEnE+1
is commutative.
Proof. Set 1n :=1∧nE, for every n∈N.
Since (1,u =Id ) is a coaugmented coalgebra and i1n is a coalgebra homomorphism, in view of
1 1 (cid:161) (cid:162) 11
Lemma 2.5, it is clear that 1∧nE,u1∧nE =i11n1 is also a coaugmented coalgebra.
Consider the following exact sequence
(2) 0 (cid:47)(cid:47)1n i11nn+1 (cid:47)(cid:47)1n+1 p11nn+1 (cid:47)(cid:47) 1n+1 (cid:47)(cid:47)0
1n
were p1n+1 denotes the canonical projection. By applying the functor 1n+1⊗(−) we get
1n
0 (cid:47)(cid:47)1n+1⊗1n 1n+1⊗i11nn+1 (cid:47)(cid:47)1n+1⊗1n+1 1n+1⊗p11nn+1 (cid:47)(cid:47)1n+1⊗ 1n+1
(cid:105)(cid:105) (cid:79)(cid:79) 1n
α1n
βn
1n+1
By Lemma 2.6, we have
(cid:181) (cid:182) (cid:104) (cid:105)
1∧ 1n
i11∧E1n ⊗ 1En ◦1ρ1∧1En1n ◦p11∧nE1n = (1∧E 1n)⊗p11n∧E1n ◦∆1∧E1n
and l−1 =1ρ so that
1∧EG 1∧EG
G (cid:181) G (cid:182) (cid:179) (cid:180)
1n+1
(3) i1n+1 ⊗ ◦l−1 ◦p1n+1 = 1n+1⊗p1n+1 ◦∆ .
11 1n 1∧EG 1n 1n 1n+1
G
We compute
(cid:181) iE (cid:182) (cid:179) (cid:180)
iE ⊗ 1n+1 ◦ 1n+1⊗p1n+1 ◦α
1n+1 1n 1n 1n+1
 (cid:179) (cid:180) (cid:179) (cid:180) 
= (cid:181)iE ⊗ iE1n+1(cid:182)◦ 1n+1⊗p11nn+1i11(cid:179)n1 ◦r1−n1+1 + (cid:180)i11n1+1 ⊗p11nn+1 ◦l1−n1+1+ 
1n+1 1n − 1n+1⊗p1n+1 ◦∆
1n 1n+1
(cid:181) iE (cid:182) (cid:104)(cid:179) (cid:180) (cid:179) (cid:180) (cid:105)
= iE ⊗ 1n+1 ◦ i1n+1 ⊗p1n+1 ◦l−1 − 1n+1⊗p1n+1 ◦∆
1n+1 1n 11 1n 1n+1 1n 1n+1
(=3)(cid:181)iE ⊗ iE1n+1(cid:182)◦(cid:183)(cid:179)i1n+1 ⊗p1n+1(cid:180)◦l−1 −(cid:181)i1n+1 ⊗ 1n+1(cid:182)◦l−1 ◦p1n+1(cid:184)=0,
1n+1 1n 11 1n 1n+1 11 1n 1∧EG 1n
G
where the last equality follows by naturality of the unit constraint.
Since iE ⊗ iE1n+1 is a monomorphism, we obtain
1n+1 1n
(cid:179) (cid:180)
(4) 1n+1⊗p1n+1 ◦α =0
1n 1n+1
so that, as the above sequence is exact, by the universal property of kernels, there exists a unique
morphism β :1n+1 →1n+1⊗1n such that
n
(cid:179) (cid:180)
(5) 1n+1⊗i1n+1 ◦β =α .
1n n 1n+1
SOME REMARKS ON CONNECTED COALGEBRAS 7
By applying the functor (−)⊗1n to (2), we get
0 (cid:47)(cid:47)1n⊗1n i11nn+1⊗1n (cid:47)(cid:47)1n+1⊗1n p11nn+1⊗1n (cid:47)(cid:47) 1n+1 ⊗1n
(cid:105)(cid:105) (cid:79)(cid:79) 1n
τn βn
1n+1
We have
(cid:181) (cid:182) (cid:179) (cid:180) (cid:179) (cid:180) (cid:179) (cid:180)
1n+1
⊗i1n+1 ◦ p1n+1 ⊗1n ◦β = p1n+1 ⊗1n+1 ◦ 1n+1⊗i1n+1 ◦β
1n 1n 1n n 1n 1n n
(cid:179) (cid:180)
(=5) p1n+1 ⊗1n+1 ◦α = 0
1n 1n+1
where the last equality can be proved similarly to (4). Since 1n+1 ⊗i1n+1 is a monomorphism we
(cid:179) (cid:180) 1n 1n
get p1n+1 ⊗1n ◦β =0 so that, as the previous sequence is exact, by the universal property of
1n n
(cid:179) (cid:180)
kernels there exists a unique morphism τ : 1n+1 → 1n ⊗1n such that i1n+1 ⊗1n ◦τ = β .
n 1n n n
Finally we have
(cid:179) (cid:180) (cid:179) (cid:180) (cid:179) (cid:180) (cid:179) (cid:180)
i1n+1 ⊗i1n+1 ◦τ = 1n+1⊗i1n+1 ◦ i1n+1 ⊗1n ◦τ = 1n+1⊗i1n+1 ◦β =α .
1n 1n n 1n 1n n 1n n 1n+1
(cid:164)
(cid:161) (cid:162)
Theorem 2.8. Let (E,∆ ,ε ),u =iE be a coaugmented coalgebra in a coabelian monoidal
E E E 1
category M. Then ε ◦i =0 and
E P(E)
(cid:179) (cid:180) (cid:161) (cid:161) (cid:162)(cid:162)
1∧E1=1∧2E,iE1∧2E = 1⊕P (E),∇ uE,iP(E) ,
(cid:161) (cid:162)
where ∇ u ,i :1⊕P (E)→E denotes the codiagonal morphism associated to u and i .
E P(E) E P(E)
Proof. Set P = P (E). Since (E,u ) is a coaugmented coalgebra, we apply Lemma 2.5 to the
E (cid:161) (cid:162)
coalgebrahomomorphismε :E →1. Thus 1,u =ε iE =Id isacoaugmentedcoalgebraand
E 1 E 1 1
α ◦ε =(ε ⊗ε )◦α . We have
1 E E E E
(6) α ◦ε ◦i =(ε ⊗ε )◦α ◦i =0.
1 E P E E E P
By definition, we have
(7) α =(1⊗u )◦r−1+(u ⊗1)◦l−1−∆ =r−1+l−1−l−1 =r−1.
1 1 1 1 1 1 1 1 1 1
Since α is an isomorphism, in view of (6), we obtain ε ◦i = 0. Consider the following exact
1 E P
sequence
iE pE E
0→1−→1 E −→1 →0.
1
Since ε ◦iE = Id , there exists a unique morphism a : E → E such that Id = iEε +apE.
E 1 1 1 E 1 E 1
Clearly the following sequence
E
0→ −a→E −ε→E 1→0.
1
is exact. From ε ◦i = 0, we get that there exists a unique morphism i(cid:48) : P → E such that
E P P 1
a◦i(cid:48) =i . Thus
P P
(cid:161) (cid:162) (cid:161) (cid:162) (cid:161) (cid:162)
∇ iE,a ◦(Id ⊕i(cid:48) )=∇ iE ◦Id ,a◦i(cid:48) =∇ iE,i .
1 1 P 1 1 P 1 P
(cid:161) (cid:162) (cid:161) (cid:162)
where ∇ iE,a :1⊕ E →E is the codiagonal morphism associated to iE and a. Since ∇ iE,a
1 1 (cid:161) 1(cid:162) 1
is an isomorphism and Id ⊕i(cid:48) is a monomorphism, we get that ∇ iE,i is a monomorphism.
1 P 1 P
Let us prove that
(cid:179) (cid:180)
α ◦ iE −iE ◦ε ◦iE =0.
E 1∧2E 1 E 1∧2E
8 A.ARDIZZONIANDC.MENINI
By Lemma 2.5 and Lemma 2.7, we have
α ◦iE =(cid:179)iE ⊗iE (cid:180)◦α =(cid:179)iE ⊗iE (cid:180)◦(cid:179)i1∧2E ⊗i1∧2E(cid:180)◦τ =(cid:161)iE ⊗iE(cid:162)◦τ
E 1∧2E 1∧2E 1∧2E 12 1∧2E 1∧2E 1 1 1 1 1 1
for a suitable τ1 :1∧2E →1⊗1. Then, by Lemma 2.5, we have
(8) (ε ⊗ε )◦α =α ◦ε (=7)r−1◦ε .
E E E 1 E 1 E
so that
(cid:161) (cid:162)
τ =(ε ⊗ε )◦ iE ⊗iE ◦τ =(ε ⊗ε )◦α ◦iE (=8)r−1◦ε ◦iE .
1 E E 1 1 1 E E E 1∧2E 1 E 1∧2E
Then
(cid:161) (cid:162) (cid:161) (cid:162)
α ◦iE = iE ⊗iE ◦τ = iE ⊗iE ◦r−1◦ε ◦iE .
E 1∧2E 1 1 1 1 1 1 E 1∧2E
On the other, by Lemma 2.5, hand we have
(cid:161) (cid:162) (cid:161) (cid:162)
α ◦iE ◦ε ◦iE = iEε ⊗iEε ◦α ◦iE (=8) iE ⊗iE ◦r−1◦ε ◦iE =α ◦iE .
E 1 E 1∧2E 1 E 1 E E 1∧2E 1 1 1 E 1∧2E E 1∧2E
(cid:179) (cid:180)
Hence αE◦ iE1∧2E −iE1 ◦εE ◦iE1∧2E =0 so that, there exists a unique morphism b:1∧2E →P such
that i ◦b=iE −iE ◦ε ◦iE . Let
P 1∧2E 1 E 1∧2E
(cid:179) (cid:180)
∆ εEiE1∧2E,b :1∧2E →1⊕P
be the diagonal morphism of εEiE1∧2E :1∧2E →1 and b:1∧2E →P. We have
(cid:161) (cid:162) (cid:179) (cid:180)
∇ iE,i ◦∆ ε iE ,b =iE ◦ε ◦iE +i ◦b=iE
1 P E 1∧2E 1 E 1∧2E P 1∧2E
so that
(cid:161) (cid:162) (cid:179) (cid:180)
(9) ∇ iE,i ◦∆ εiE ,b =iE .
1 P 1∧2E 1∧2E
We have
(cid:161) (cid:162) (cid:161) (cid:162)
pE ⊗pE ◦∆ ◦∇ iE,i
1 1 E 1 P
(cid:163)(cid:161) (cid:162) (cid:161) (cid:162) (cid:164)
= ∇ pE ⊗pE ◦∆ ◦iE, pE ⊗pE ◦∆ ◦i
1 1 E 1 1 1 E P
(cid:169)(cid:161) (cid:162) (cid:161) (cid:162) (cid:161) (cid:162) (cid:163)(cid:161) (cid:162) (cid:161) (cid:162) (cid:164)(cid:170)
= ∇ pE ⊗pE ◦ iE ⊗iE ◦∆ , pE ⊗pE ◦ E⊗iE ◦r−1+ iE ⊗E ◦l−1 =0
1 1 1 1 E 1 1 1 E 1 E
(cid:161) (cid:162)
so that there exists a unique morphism Γ iE1,iP :1⊕P →1∧2E such that
(cid:161) (cid:162) (cid:161) (cid:162)
∇ iE,i =iE ◦Γ iE,i .
1 P 1∧2E 1 P
(cid:161) (cid:162) (cid:161) (cid:162)
Since ∇ iE,i is a monomorphism, so is Γ iE,i . On the other hand, we get
1 P 1 P
(cid:161) (cid:162) (cid:179) (cid:180)
iE ◦Γ iE,i ◦∆ εiE ,b (=9)iE .
1∧2E 1 P 1∧2E 1∧2E
(cid:161) (cid:162) (cid:179) (cid:180) (cid:161) (cid:162)
Since iE is a monomorphism, we get Γ iE,i ◦∆ εiE ,b = Id and hence Γ iE,i is
1∧2E (cid:161) (cid:162) 1(cid:179)P (cid:180) 1∧2E 1∧2E 1 P
also an epimorphism. Thus Γ iE,i and ∆ εiE ,b are mutual inverses. (cid:164)
1 P 1∧2E
Definition 2.9. A graded coalgebra (C = ⊕ C ,∆ ,ε ) in a cocomplete monoidal category
n∈N n C C
M is called 0-connected whenever εC =ε iC :C →1 is an isomorphism
0 C 0 0
A graded coalgebra C in a cocomplete monoidal category M is called strictly graded whenever
1) C is 0-connected;
2) C is a strongly N-graded coalgebra.
Next theorem provides our main example of a strictly graded coalgebra.
SOME REMARKS ON CONNECTED COALGEBRAS 9
Theorem 2.10. Let M be a cocomplete coabelian monoidal category such that the tensor product
commutes with direct sums. Let ((C,∆,ε),u ) be a coaugmented coalgebra in M.
C
Then the associated graded coalgebra
1∧ 1 1∧ 1∧ 1
gr C =1⊕ C ⊕ C C ⊕···
1 1 1∧ 1
C
is a strictly graded coalgebra.
(cid:179) (cid:180)
Proof. By[AM2,Theorem2.10],wehavethat gr C,∆ ,ε =ε ◦u ◦pgr1C isastrongly
1 gr1C gr1C C C 0
N-graded coalgebra. Since u is a coalgebra homomorphism, we get
C
ε =ε◦u ◦pgr1C =pgr1C.
gr1C C 0 0
It is now clear that εgr1C := ε ◦igr1C = Id so that gr C also 0-connected and hence it is a
0 gr1C 0 1 1
strictly graded coalgebra. (cid:164)
Theorem 2.11. Let (C = ⊕ C ,∆ ,ε ) be a 0-connected graded coalgebra in a cocomplete
n∈N n C C
coabelian monoidal category M (e.g. C is strictly graded). Then
(cid:161) (cid:162)
1) (C,∆ ,ε ),u =iC is a connected coalgebra where u :=iCε−1 :1→C.
(cid:181) C C(cid:182) C 1 C 0 0
(cid:161) (cid:161) (cid:162)(cid:162) (cid:161) (cid:162)
2) C∧2C,iC = C ⊕P (C),∇ iC,i , where ∇ iC,i :C ⊕P (C)→C denotes
0 C∧2C 0 0 P(C) 0 P(C) 0
0
the diagonal morphism associated to iC and i .
0 P(C)
Moreover if M is also complete and satisfies AB5, then the following assertions are equivalent:
(a) C is a strongly N-graded coalgebra.
(cid:161) (cid:162) (cid:161) (cid:162)
(b) C ,iC = P (C),i .
1 1 P(C)
In particular, when (b) holds, C is a strictly graded coalgebra.
(cid:161) (cid:162)
Proof. 1) By Proposition [AM1, Proposition 2.5], C ,∆ =∆ ,ε =εiC is a coalgebra in M
0 0 0,0 0 0
andiC isacoalgebrahomomorphism. Henceε andiC arebothcoalgebrahomomorphismssothat
0 0 0
δ :=iCε−1 is a coalgebra homomorphism and hence ((C,∆ ,ε ),δ) is a coaugmented coalgebra.
0 0 C C
∧t
By [AMS1, Proposition 3.3], we have C = lim(C C) . Since ε is a coalgebra isomorphism, we
−→ 0 t∈N 0
conclude that −li→m(1∧nC)n∈N =C i.e. that ((C,∆C,εC),δ) is a connected coalgebra.
2) It follows by 1) and in view of Theorem 2.8.
Now, assume that M is also complete and satisfies AB5 and let us prove that (a) and (b) are
equivalent. By 2), we have
(cid:179) (cid:180) (cid:161) (cid:161) (cid:162)(cid:162)
C∧2C,δ = C ⊕P (C),∇ iC,i .
0 2 0 0 P(C)
Then, by [AM1, Theorem 2.22], (a) is equivalent to
(cid:161) (cid:161) (cid:162)(cid:162) (cid:179) (cid:180)
C ⊕C ,∇ iC,iC = C∧2C,δ
0 1 0 1 0 2
and hence to
(cid:161) (cid:161) (cid:162)(cid:162) (cid:161) (cid:161) (cid:162)(cid:162)
(10) C ⊕C ,∇ iC,iC = C ⊕P (C),∇ iC,i .
0 1 0 1 0 0 P(C)
(b)⇒(10) It is trivial.
(10) ⇒ (b) By hypothesis there exists an isomorphism Λ : C ⊕ P (C) → C ⊕ C such that
(cid:179) (cid:180) (cid:161) (cid:162) 0 0 1
∇ iC,iC =∇ iC,iC ◦Λ.
0 P(C) 0 1
Let πC0⊕C1 :C ⊕C →C be the canonical projection for a=0,1. We have
Ca 0(cid:161) 1 (cid:162) a (cid:161) (cid:162) (cid:161) (cid:162)
ε ◦∇ iC,iC =∇ ε ◦iC,ε ◦iC =∇ ε ,0 =ε ◦πC0⊕C1
C 0 1 C 0 C 1 0 Hom(C1,1) 0 C0
and by Theorem 2.8, we have
(cid:161) (cid:162) (cid:179) (cid:180) (cid:161) (cid:162)
ε ◦∇ iC,i =∇ ε ◦iC,ε ◦iC =∇ ε ,0 =ε ◦πC0⊕P(C).
C 0 P(C) C 0 C P(C) 0 Hom(P(C),1) 0 C0
Hence, by definition of Λ we get
(cid:161) (cid:162) (cid:161) (cid:162)
ε ◦πC0⊕C1 ◦Λ=ε ◦∇ iC,iC ◦Λ=ε ◦∇ iC,i =ε ◦πC0⊕P(C).
0 C0 C 0 1 C 0 P(C) 0 C0
10 A.ARDIZZONIANDC.MENINI
Since ε is an isomorphism, we get that
0
(11) πC0⊕C1 ◦Λ=πC0⊕P(C).
C0 C0
Consider the following diagram
iC0⊕P(C) πC0⊕P(C)
0 (cid:47)(cid:47)P(C) P(C) (cid:47)(cid:47)C ⊕P (C) C0 (cid:47)(cid:47)C (cid:47)(cid:47)0
0 0
b Λ IdC0
(cid:178)(cid:178) iC0⊕C1 (cid:178)(cid:178) πC0⊕C1 (cid:178)(cid:178)
0 (cid:47)(cid:47)C C1 (cid:47)(cid:47)C ⊕C C0 (cid:47)(cid:47)C (cid:47)(cid:47)0
1 0 1 0
where the rows are exact and the right square commutes. Hence there is a unique morphism
b:P (C)→C such that the left square commutes too. Clearly b is an isomorphism. Moreover
1
(cid:161) (cid:162) (cid:161) (cid:162) (cid:179) (cid:180)
iC ◦b=∇ iC,iC ◦iC0⊕C1 ◦b=∇ iC,iC ◦Λ◦iC0⊕P(C) =∇ iC,iC ◦iC0⊕P(C) =iC
1 0 1 C1 0 1 P(C) 0 P(C) P(C) P(C)
(cid:161) (cid:162) (cid:179) (cid:180)
so that C ,iC = P (C),iC . (cid:164)
1 1 P(C)
Remark 2.12. Let (C = ⊕ C ,∆ ,ε ) be a graded coalgebra in a cocomplete and complete
n∈N n C C
coabelian monoidal category M satisfying AB5. In view of Theorem 2.11, C is strictly graded if
and only if it is 0-connected and
(cid:161) (cid:162) (cid:161) (cid:162)
C ,iC = P (C),i .
1 1 P(C)
Note that, when M is the category of vector spaces over a field K, our definition agrees with
Sweedler’s one in [Sw, page 232].
Theorem2.13. Let(B,m ,u ,∆ ,ε )beabraidedgradedbialgebrainacocompleteandcomplete
B B B B
braided monoidal category (M,c) such that M is abelian satisfying AB5. Assume that the tensor
products are additive, commute with direct sums and are (two-sided) exact. Assume that B is
0-connected as a coalgebra.
Then ((B,∆ ,ε ),u ) is a connected coalgebra.
B B B
Moreover B is the braided bialgebra of type one B [B ] associated to B and B if and only if
0 1 0 1
(⊕ B )2 =⊕ B and P (B)=B .
n≥1 n n≥2 n 1
Proof. By[AM1,Theorem6.10]andTheorem2.11,((B,∆ ,ε ),δ)isaconnectedcoalgebrawhere
B B
δ :=iBε−1 :1→B. Moreover B is the braided bialgebra of type one B [B ] associated to B and
0 0 0 1 0
B if and only if (⊕ B )2 =⊕ B and P (B)=B .
1 n≥1 n n≥2 n 1
Let us prove that δ =u .
B
By [AM1, Propositions 2.5 and 3.4], ε =ε ◦pB and u =iB ◦u where u =pB ◦u . Thus
B 0 0 B 0 0 0 0 B
Id =ε ◦u =ε ◦pB ◦iB ◦u =ε ◦u
1 B B 0 0 0 0 0 0
and hence u =ε−1. Then we get u =iB ◦u =iB ◦ε−1. (cid:164)
0 0 B 0 0 0 0
Remark 2.14. RecallthataTOBA(alsocalledbraidedHopfalgebraoftypeone)[AG,Definition
3.2.3] in the category HYD of Yetter Drinfeld modules over an ordinary K-Hopf algebra H is a
H
graded bialgebra T =⊕ T in this category such that
n∈N n
T (cid:39)K, (⊕ T )2 =⊕ T , and P (T)=T .
0 n≥1 n n≥2 n 1
Therefore TOBAs are exactly the bialgebras described in Theorem 2.13 in the case when M =
HYD.
H