SOME REMARKS ON MORITA THEORY, AZUMAYA ALGEBRAS AND CENTER OF AN ALGEBRA IN BRAIDED
MONOIDAL CATEGORIES
B. FEMI ´C
Abstract. We study the left and the right center of an algebra in a braided monoidal categoryC. We characterize them by specializing a general result and point out why in symmetric monoidal categories one does not distinguish the two centers. We review Morita theorems for a monoidal categoryDand analyze necessary conditions for the associativity of the tensor product over an algebra inD. The central part of the Morita theorems in braided categories is also discussed. We review the notion of Azumaya algebra inCdistinguishing left and right faithfully projective objects and study the relation between the two.
1. Introduction
The well known Morita Theorems address the question of the equivalence of categories of modules over algebras. The induced equivalence relation between algebras in the case of Azumaya algebras underlies the construction of the Brauer group. Pareigis generalized the Morita Theorems to arbitrary monoidal categories in [9] and the construction of the Brauer group to symmetric monoidal categories in [10]. The latter was a big step in the line of generalizations of the Brauer group. This group was initially considered for the category of vector spaces in order to classify finite dimensional central division algebras and its computation is related to number theory, algebraic geometry andK-Theory. This concept reaches its highest level of generalization in [11] – in the context of braided monoidal categories. For an overview of different constructions of Brauer groups see [1].
In the present paper we review (relying on [9] and [11]) the Morita Theorems in monoidal categories as well as the notion of Azumaya algebra analyzing some issues whose close study was omitted in the mentioned references. We prove that two Morita equivalent algebras in a braided monoidal categoryC have isomorphic centers and that Azumaya algebras are central, generalizing the results from [10].
The center of an algebra A in a symmetric monoidal category was introduced in [9, 10] as the inner hom-objectA[A, A]A. We study this object inC and prove
not impose the separability condition onA(nor abelianity ofC), and we show that the isomorphism in question preserves the algebra structure. We explain why in symmetric categories one does not distinguish the left and the right center of an algebra, as one does in braided categories.
Developing Morita Theory in a monoidal categoryC, Pareigis defined in [9] the concept of faithfully projective object inAC, whereA∈ Cis an algebra.
Symmetri-cally, in [11] the authors study the notion of faithfully projective object inCA and
work with the Morita Theorems from [9] together with their right hand-side coun-terparts. ForA=I, whereI∈ C is the unit object, an object faithfully projective in
IC (respectively inCI) we call left (respectively right) faithfully projective object.
In the construction of the Brauer group of a symmetric monoidal category in [10], an Azumaya algebra is an algebraA which is (left) faithfully projective and such that there is a certain algebra isomorphismF:A⊗Aop−→[A, A]. When extending the construction of the Brauer group from symmetric to general braided monoidal categories in [11], besideF there appears another algebra isomorphismG:Aop⊗A −→ [A, A]op (in symmetric monoidal categories F and G coincide, as we show in Section5). Though, it is not made explicit which kind of faithful projectiveness is used.
We verify in Theorem 5.1 that an Azumaya algebraA in a braided monoidal category is both left and right faithfully projective. More precisely, in [11, Theorem 3.1] the authors characterize Azumaya algebras in terms of two equivalence functors
F andG, without providing an explicit proof. We prove that the functorF gives rise to the algebra isomorphismF and it implies thatAis right faithfully projective, whereas the algebra isomorphismGand the left faithful projectiveness ofA come out from the functorG. We also show that the two equivalence functorsF andG
coincide in a symmetric categoryDfor the same reason for which the left and the right center of an algebra are not distinguished inD.
Moreover, we address the issue of the relation between the notions of left and right faithfully projective objects in a braided monoidal category and prove the equivalence of the two notions. In view of this, the statement of [11, Theorem 3.1] remains valid.
For the Morita Theorems in [9] one needs the associativity of the tensor prod-uct over an algebra. We investigate necessary conditions for this associativity in Lemma 4.3. Our results lead to assumptions in Morita Theorems which slightly differ from those in [9].
The contents of the paper are organized as follows. In the second section we recall some definitions and basic facts for a braided monoidal categoryC. In Section3we study and characterize the left and the right center of an algebra inC. Section4
Acknowledgements. This work has been partially developed in the Mathe-matical Institute of the Serbian Academy of Sciences and Arts in Belgrade (Ser-bia). The author wishes to thank to Facultad de Ciencias de la Universidad de la Rep´ublica in Montevideo for their warm hospitality and provision of the necessary facilities.
2. Preliminaries
We assume the reader is familiar with the general theory of (braided monoidal) categories. For reference we recommend [7,4]. ThroughoutCwill denote a braided monoidal category (with braiding Φ and unitI) unless otherwise specified. We set the notation for the structures of a (commutative) algebra A, an A-moduleM, a morphismf :X −→Y inC and for the braiding ΦX,Y forX, Y ∈ C:
Algebra Module Braiding Morphism
unit multiplic. comm. alg. left right
ηA=r
A
∇A=
A A
A A A
A =
A A
A
νM=
A M
PP
M
µM=
M A
M
X Y
Y X
X
f
Y
2.1. Observe that for algebrasA, Bin a monoidal categoryCthe categoriesACof
leftA-modules andCBof rightB-modules are right and left module categories over Crespectively. For the definition of a module category over a monoidal category and their functors we refer to [6]. For brevity, left and right module categories overCwill be calledleft and right C-categoriesand the corresponding strict module functors we will call shortlyC-functors. A natural transformationϕ:F −→ G between two right C-functors F,G : AC −→BC is called a right C-natural transformation if for
allM ∈AC andX ∈ C the following diagram commutes
F(M⊗X) -G(M⊗X)
ϕ(M⊗X)
?
∼ =
?
∼ =
F(M)⊗X -G(M)⊗X. ϕ(M)⊗X
Similarly, one defines a leftC-natural transformation.
When C is braided we can considerAC as a left andCB as a rightC-category
respectively with the corresponding structures:
A X⊗M
PP
X⊗M =
A X M
PP
X M
(1) and
N⊗X B
N⊗X =
N X B
N X
(2)
forM ∈AC, N∈ CBandX ∈ C. The category ofA-B-bimodulesACBis isomorphic
(as a rightC-category) toA⊗BC through correspondences
A⊗B M
PP
M =
A B M
PP
M
; (3) A NP
P
N =
A N
r
A⊗B
PP
N and
N B
N =
N B
r
A⊗B
PP
ForK∈ACB andX∈ Cthe objectK⊗X inherits its leftA-module structure from
that onK, whereas its right B-module structure is given as in (2). Analogously,
ACB is isomorphic (as a leftC-category) toCA⊗B via
M A⊗B
M =
M A B
PP
M
; (4) A N
PP
N =
A N
r
A⊗B
N
and N B
N =
N B
r
A⊗B
N
whereACB is considered a leftC-category through (1). Hence,A⊗BCis isomorphic
toCA⊗B. However, this is neither left nor rightC-isomorphism of categories,unless the categoryC is symmetric.
In Sections 4 and 5 both rightC-functorsAC −→BCand leftC-functorsCA−→ CB
we will call shortlyC-functors.
2.2. A monoidal categoryCis calledright closedif the functor−⊗M :C → Chas a right adjoint, denoted by [M,−] and calledinner hom-functor, for allM ∈ C. For
N ∈ C, the object [M, N] is called inner hom-object. The counit of the adjunction evaluated atN is denoted byevM,N : [M,N]⊗M →N. The pair ([M, N],evM,N)
satisfies the following universal property: for any morphismf :T⊗M →N there is a unique morphismg:T →[M, N]such thatf =evM,N(g⊗M). Iff :N−→N0 is
a morphism inC, then [M, f] : [M, N]−→[M, N0] is the unique morphism such that
evM,N0([M,f]⊗M) =f evM,N. The unit of the adjunctionα:N −→[M, N⊗M]
is induced byevM,N⊗M(α⊗M) =idN⊗M.
A monoidal category C is called left closed if the functor M ⊗ − : C → C has a right adjoint {M,−} for allM ∈ C. The counit of this adjunction evaluated at
N ∈ Cis denoted byevM,N :M⊗ {M, N} →N. The properties analogous to those
in the previous paragraph hold for{M, N}. We denote the unit of the adjunction (M ⊗ −,{M,−}) by α : N −→ {M, M ⊗N}. The category C is called closed if it is both left and right closed. When there is no confusion, we will suppress the indexes for the counitev.
WhenCis braided we have a natural isomorphismC(M⊗P, N)−→ C(P⊗M, N), f 7→f ◦(ΦP,M)±1. This clearly implies that C is right closed if and only if it is
left closed. By uniqueness of the (right) adjoint functors, the functors {M,−}
and [M,−] are naturally equivalent, then for all N ∈ C it is {M, N} ∼= [M, N]. For the counits of the two adjunctions, putting P = [M, N], we obtain ev =
ev◦(Φ[M,N],M)∓1. Throughout we will write [−,−] for both types of inner
hom-bifunctors, unless otherwise specified. The difference will be clear from the context.
2.3. Let P be an object in a monoidal category C. An object P∗∈ C together with a morphismeP:P∗⊗P −→I is called a left dual object forP if there exists
a morphism dP: I −→P⊗P∗ in C such that (P ⊗eP)(dP ⊗P)= idP and (eP ⊗ P∗)(P∗⊗dP)=idP∗. In braided diagrams the morphismseP anddP are denoted
by:
eP = P∗ P and dP =
Symmetrically, one defines a right dual object. Dual objects are unique up to isomorphism. In a braided category left and right dual objects are isomorphic.
2.4. On inner hom-objects in a right closed monoidal category there is an asso-ciative pre-multiplicationϕM,Y,Z : [Y, Z]⊗[M, Y]−→[M, Z] and a unital morphism η[Y,Y]:I−→[Y, Y] such that
ϕM,Y,Y(η[Y,Y]⊗[M, Y]) =λ[M,Y] and ϕM,M,Y([M, Y]⊗η[M,M]) =ρ[M,Y]
where λ and ρ are unity constraints. The morphisms ϕM,Y,Z and η[Y,Y] are
given via the universal properties of ([M, Z], ev) and ([Y, Y], ev), respectively, by
evM,Z(ϕM,Y,Z⊗M) =evY,Z([Y,Z]⊗evM,Y) andevY,Y(η[Y,Y]⊗Y) =λY.
Con-sequently, for eachM ∈ C, the object [M, M] is equipped with an algebra structure viaϕM,M,Mandη[M,M]. The analogous statement we have for the algebra{M, M}.
2.5. For any algebraAin a braided monoidal categoryCwe adopt the convention to consider the multiplication on the opposite algebraA to be given by∇AΦA,A.
This implies that from the two possibilities for the relation between the counits of the adjunctions (see2.2) one has to consider ev = evΦ. Indeed, observe that by the left version of the multiplication in2.4we have
M{M,M} {M,M}
h
ev
M
=
M{M,M} {M,M}
h
ev
h
ev
M
which is equivalent to
{M,M} {M,M}M
ev
M
=
{M,M} {M,M}M
ev
ev
M
Choosingev =evΦ and using an isomorphismδ:{M, M} −→[M, M] in order to pass from{M, M}to [M, M], we further obtain
{M,M} {M,M}M
δ
h
ev
M
=
{M,M} {M,M}M
δ δ ev
h
ev
M
=
{M,M} {M,M}M
δ δ
h
ev
M
which recovers the known algebra isomorphism{M, M} ∼= [M, M] ([11, Corollary 2.2]). If we had chosen the other relation betweenev andev, we would have to im-pose symmetricity conditions on the braiding (Φ{M,M},{M,M}= (Φ{M,M},{M,M})−1) in order to recover the mentioned isomorphism.
2.6. Let C be a monoidal category with coequalizers. Atensor product over an algebraAinC of a rightA-moduleM and a leftA-moduleN is the coequalizer
M⊗AN.
-ΠM,N
-µM⊗N
M⊗A⊗N -M⊗N
M⊗νN
Consider a right A-linear morphism f : M −→ M0 and a left A-linear morphism
3. The left and the right center of an algebra
The center of an algebra in a symmetric monoidal category was defined in [10]. If a category is braided, one distinguishes two counterparts of the center of an algebra. This was first recognized in [11] where the notion of the left and the right center was introduced for a separable algebra in an abelian braided monoidal category. In [6] this notion was studied for an algebra in a semisimple abelian ribbon category, while [3] treats it in terms of a maximality property for idempotents of symmetric special Frobenius algebras. In this section we study the left and the right center of an algebra A in a closed braided monoidal category C. We work in a closed category in order to guarantee the existence of inner hom-objects. Before defining the left and the right center ofA, we recall some concepts.
Take M, N ∈AC and let A[M, N] be the inner hom-object obtained from the
adjunctionM ⊗ −:C −→AC. Pareigis proved in [8] thatA[M, N] is the equalizer
A[M, N] λ-[M, N]
-l1
[A⊗M, N],
-l2
(5)
wherel1, l2: [M, N]−→[A⊗M, N] are induced byev(A⊗M⊗l1)=ev(νM⊗[M, N])
andev(A⊗M ⊗l2)=νN(A⊗evM,N). Then (5) yields:
ev(νM⊗λ) =νN(A⊗ev(M⊗λ)). (6)
The functorA[M,−] acts on morphisms as follows. If f:N−→N0 is a morphism
in AC, then A[M, f]:A[M, N]−→A[M, N0] is induced by [M, f]: [M, N]−→[M, N0]
as a morphism on an equalizer viaλN0◦A[M, f] = [M, f]λN. That [M, f] induces A[M, f] is provided byA-linearity off. Analogously, for M, N ∈ CA considering
the right adjoint functor for− ⊗M :C −→ CA, one gets that
[M, N]A -[M, N]
ρ
-r1
[M⊗A, N]
-r2
(7)
is the equalizer, withr1, r2 : [M, N]−→[M ⊗A, N] induced byev(r1 ⊗M ⊗A)= ev([M,N]⊗µM) andev(r2 ⊗M⊗A)=µN(evM,N⊗A). A morphismf :N −→N0
inCA induces a morphism [M, f]A: [M, N]A−→[M, N0]A.
For another algebraB ∈ CandM∈ACB, consider the functors: M⊗−,−⊗M :C −→ACB with their right adjoint A[M,−]B. For X∈C we viewM⊗X andX⊗M as A-B-bimodules as in2.1. Pareigis defined in [10] the center of an algebraA in a
symmetric monoidal category as the object A[A, A]A (if it exists). We will study
the properties of this object inC further below.
Definition 3.1. The left (right) center of an algebraAin a closed braided monoidal categoryC is the object A⊗A[A, A](resp. [A, A]A⊗A) inC.
The algebraA⊗Ais calledthe enveloping algebra ofAand it is denoted byAe.
The algebraA⊗Awe denote byAe. We regardAas a leftAe-module via (3) and
as a right Ae-module via (4). We give another interpretation of the left and the
Definition 3.2. ForM ∈AeC, letMA denote the equalizer:
MA -j M α-A [A, A⊗M]
-[A, A⊗ηA⊗M]
[A, A⊗A⊗M]
-[A, ηA⊗A⊗M]
[A, M],
-[A, ν]
whereαAis the unit of the adjunction(A⊗−,[A,−])andν the structure morphism forM.
Note that ν(A⊗ηA⊗M) =µl and ν(ηA⊗A⊗M) =µrΦA,M by (3), where µl and µr stand for the left- and the right A-action on M respectively. Using
the identities from2.2we compute ev(A⊗[A, θ]αA) = θev(A⊗αA) = θ for any θ:A⊗M −→M. Substitutingθ byµl andµrΦA,M, we obtain a short description
of the property satisfied by the equalizerMA:
A MA
j
PP
M =
A MA
j
M.
(8)
If f : M −→ N is a morphism in AeC, then fA : MA −→ NA is induced as a
morphism on an equalizer viajNfA=f jM.
The following result is proved in author’s Ph.D. thesis.
Proposition 3.3. LetAbe an algebra inC. Then the functorsAe[A,−]and(−)A are isomorphic.
Proof. Take M ∈ AeC. We define the morphisms g : [A, M] −→ M and h : M −→ [A, M] by g := ev(ηA ⊗[A, M]) and ev(A⊗h) = µl. We view A as a left Ae-module by (3). In the diagram
?
g 6
h
Ae[A, M] λ-[A, M]
MA -M
j ? g 6 h -l1
[(A⊗A)⊗A, M]
-l2
-µl
[A, M]
-µrΦA,M 1
we are going to prove that the compositiongλ inducesgandhj induceshso that the squareh1icommutes both with left and right arrows. We proceed withg:
A Ae[A,M]
λ g
PP
M
g (3)
=
A Ae[A,M]
r r λ A⊗A
h
ev
PP
M
(6)
=
A Ae[A,M]
r r
A⊗A λ
PP
h
ev
M
? (3)
= A Ae[A,M]
λ
h
ev
M
(3)
=
A Ae[A,M]
r r
A⊗A λ
PP h
ev
M
(6)
=
A Ae[A,M]
r r λ A⊗A
h
ev
PP
M
g (3)
= A Ae[A,M]
λ g
By the equalizer property of MA this means that gλ induces g so that jg =gλ.
Let us now prove thathis well defined. We compute:
A⊗A A MA
j
PP h
h
ev
M
h (3)
=
A A A MA
j PP M ass. mod. =
A A A MA
j
PP
PP
M
(8) mod.
=
A A A MA
j PP PP M bimod. =
A A A MA
j PP PP M nat. (3)
=
A⊗A A MA
j PP PP M h =
A⊗A A MA
j h h ev PP M
Thus hj induces hso that λh =hj. We next prove that g and h are inverse to each other. We have: ev(A⊗hgλ) = µl(A⊗gλ) =ev(A⊗λ), where the latter
equality holds by the identity ?. This yields λ=hgλ=hjg=λhg. Sinceλ is a monomorphism, we gethg=idAe[A,M]. On the other hand, it isgh=ev(ηA⊗h) = µl(ηA⊗M) =idM. Composing this from the right withj and using the fact that
it is a monomorphism, similarly as above we obtain gh=idMA.
We finally prove that the isomorphismAe[A, M]∼=MA is natural. LetN∈AeC
and letf :M −→N be a morphism inAeC. Observe the following diagram
Ae[A, M] -[A, M]
λM
?
Ae[A, f]
Ae[A, N]
M? gM
N.? f
[A, N?]
λN -gN H H H HHj
H H
H HHj
gM
gN
MA
N?A
fA HH H
HHj
H H
H HHj
jM
jN
The upper and the lower triangle in this picture commute by the definitions ofgM
and gN, respectively. The right inner trapeze commutes by the definition of fA.
The outer diagram commutes as well, it can be seen as a juxtaposition
?
Ae[A, f]
Ae[A, M] λM-[A, M]
Ae[A, N] -[A, N]
λN ? [A, f]
M -gM ? f N -gN
where the left inner parallelogram commutes by the definition of the morphism
Ae[A, f], and the right one by the definitions ofgand [A, f]. Now a diagram
chas-ing argument applied to the previous diagram provides jNgN Ae[A, f] =jNfAgM,
which, sincejN is a monomorphism, yields thatg is a natural transformation.
Symmetrically toMAone definesAM forM∈ C
Ae using the adjunction (− ⊗A,
[A,−]). The short description of its equalizer property takes the form:
AM A ˜ j M =
AM A ˜
j
PP
M.
Similarly as in Proposition 3.3 one has that [A,−]Ae and A(−) are isomorphic
functors.
Corollary 3.4. We have the following isomorphisms of commutative algebras:
A⊗A[A, A]∼=A
A and [A, A] A⊗A∼=
AA.
Proof. Using multiplication∇[A,A] in [A, A] (see2.4) and associativity of∇A, one
proves that the morphism h : A −→ [A, A] from the above proof is an algebra morphism. We define ∇: Ae[A, A]⊗Ae[A, A]−→ Ae[A, A] by λ∇=∇[A,A](λ⊗λ).
Then Ae[A, A] is a subalgebra of [A, A] and λ : Ae[A, A] −→ [A, A] is an algebra
morphism. Similarly,j :AA −→A is an algebra embedding. Then λ∇(h⊗h) = ∇[A,A](λh⊗λh)
h1i
= ∇[A,A](hj⊗hj) =h∇A(j⊗j) =hj∇AA
h1i
= λh∇AA, whereh1i
denotes the square diagram in the proof of Proposition3.3. Sinceλis a monomor-phism, the latter yields that h : AA −→ Ae[A, A] is an isomorphism of algebras.
To prove commutativity, compose the identity (8) withj⊗AA from above (in the
braided diagram orientation), settingM =A. Sincejis an algebra morphism and a monomorphism, we obtain thatAA is commutative. The other isomorphism is
proved similarly.
The isomorphisms in the above Corollary were proved in [11, Proposition 4.4] on the level of objects for separable algebras (in abelian categories). We obtained our result as a special case of Proposition3.3 and its right hand-side version. In view of (8) and (9) our left and right centers coincide with right and left centers of an algebra in a semisimple abelian ribbon category from [6].
3.1. The objectA[A, A]A. Before we compare objectsA⊗A[A, A], [A, A]A⊗Aand
A[A, A]Ain a symmetric category, let us study the latter object in a closed braided
monoidal category C. Being an inner hom-object, A[A, A]A is the representing
object of the two functorsACA(A⊗ −, A),ACA(− ⊗A, A) :C −→ Set, where Set
is the category of sets. Similarly as we saw that the inner hom-objects A[M, N]
and [M, N]B are equalizers, we will prove that the inner hom-objectA[M, N]B is
a (categorical) limit. This will provide us with some identities relatingA[A, A]A
with objectsA[A, A],[A, A]A and [A, A].
Let us first remark that for A-B-bimodulesM and N the set of A-B-bimodule morphismsACB(M, N) can be described using (braided) diagrams in the (braided
monoidal and closed) category of sets as follows:
ACB(M,N)A M B j PP
h
ev
N
=
ACB(M,N)A M B j
h
ev
PP
N
=
ACB(M,N)A M B
j
h
ev
PP
N =
ACB(M,N)A M B j PP
h
ev
N
=
ACB(M,N) A⊗M⊗B j ui
h
ev
N
(C(A⊗M⊗B, N),ev) as the first, second, third and fourth diagram, respectively, in (10) show (omitting the morphism j). We use the same notation ev for both counit inC and inSet. In Setthis means thatACB(M, N) is the limit of theui’s. Lemma 3.5. LetM andN beA-B-bimodules and suppose thatCis complete. The objectA[M, N]B is the limit
A[M, N]B ι-[M, N]
-v1
-v2
-v3
-v4
[A⊗M⊗B, N], (11)
wherev1, v2, v3, v4 are given by the universal property of([A⊗M ⊗B, N],ev) via the equality of the first diagram with the second, third, fourth and fifth diagram in the below picture, respectively:
[M,N]A⊗M⊗B vi
h
ev
N :
[M,N]A M B
PP
h
ev
N ;
[M,N]A M B
h
ev
PP
N ;
[M,N]A M B
h
ev
PP
N ;
[M,N]A M B
PP
h
ev
N.
(12)
Proof. We define morphismsu0i :C(X⊗M, N)−→ C(A⊗X⊗M⊗B, N), i= 1,2,3,4, as the compositions
u0i: C(X⊗M, N) u-i C(A⊗X⊗M⊗B, N) -C(X⊗A⊗M⊗B, N) C(Φ−A,X1 ⊗M⊗B, N)
being ui’s the morphisms we defined in (10), where M is substituted by X⊗M.
LetL(M, N) be the limit of the morphisms v1, v2, v3, v4 from the Lemma. In the
diagram:
ACB(X⊗M, N) -C(X⊗M, N)
-u01
-u02
-u0 3
-u0 4
C(X⊗A⊗M⊗B, N)
C(X, L(M, N)) -C(X,[M, N])
-C(X, v1)
-C(X, v2)
-C(X, v3)
-C(X, v4)
C(X,[A⊗M⊗B, N])
?
∼ =
?
∼ =
the first row is a limit, because C(Φ−A,X1 ⊗M⊗B, N) is an isomorphism, and the second row is a limit too, because C(X,−) as a covariant representable functor preserves limits. Direct check shows that the right rectangle commutes taking into account the four pairs of arrows (u0i,C(X, vi)). From the isomorphism of the two
limits we obtain thatL(M,−) is a right adjoint for the functor− ⊗M :C −→ACB,
henceL(M, N)∼=A[M, N]B and we have the claim.
impliesev(A⊗M⊗l1ι) =ev(A⊗M⊗l2ι) because of (11). The universal property
of ([A⊗M, N],ev) yields l1ι =l2ι, hence by the equalizer property of A[M, N],
the morphismι :A[M, N]B −→[M, N] factors through a morphismιl :A[M, N]B −→A[M, N] making the triangle
A[M, N] λ-[M, N] A[M, N]B
?
ιl H
H H
HHj
ι
(13)
commutative. Similarly, we obtain thatι:A[M, N]B −→[M, N] factors through a
morphismιr:A[M, N]B −→[M, N]B so that the triangle
[M, N]B ρ-[M, N] A[M, N]B
?
ιr H
H H
HHj
ι
(14)
commutes.
On the other hand, for the left centerAe[A, A] ofA, from (6) and (3) one obtains
the identity
A A AAe[A,A]
λe
h
ev
A =
A A A Ae[A,A]
λe
h
ev
A
After composing the latter equality from above with A⊗ηA⊗A⊗id, one gets
that λe : Ae[A, A]−→ [A, A] factors through a morphismλl : Ae[A, A] −→A[A, A]
so thatλλl=λe. Similarly, one proves thatρe: [A, A]Ae −→[A, A] factors through
a morphismρr : [A, A]Ae −→[A, A]A so thatρρr=ρe. Here λ:A[A, A]−→[A, A]
andρ: [A, A]A−→[A, A] denote the corresponding equalizer monomorphisms.
Lemma 3.6. For X ∈ AC it is A[A, X] ∼= X. Symmetrically, for Y ∈ CA it is
[A, Y]A∼=Y.
Proof. Definef :X −→[A, X] byev(A⊗f) =νX. It inducesf :X−→A[A, X] such
thatλf =f, since: ev(∇A⊗f) =νX(∇A⊗X) =νX(A⊗νX) =νX(A⊗ev(A⊗f)).
Now the morphism ζ := ev(ηA ⊗[A, X])λ : A[A, X] −→ X is the inverse of f.
Indeed,ζf =ev(ηA⊗λf) =ev(ηA⊗f) =νX(ηA⊗X) =idX andev(A⊗(λf ζ))= ev(A⊗fev(ηA⊗λ))=νX(A⊗ev(ηA⊗λ))=ev(∇A(A⊗ηA)⊗λ)=ev(A⊗λ).In the
penultimate equality we applied (6). From the universal property of ([A, X],ev) it follows λf ζ =λ. Since λis a monomorphism, we get f ζ = idA[A,X]. For the
second claim one proves thatg :Y −→[A, Y] defined by ev(g⊗A) = µY induces
Consider the diagrams
Ae[A, A]
?
λl λe
H H
H H
H H j
A[A, A]A ιl -A[A, A] λ -[A, A]
f
*
A
6
f ιL
H H
H H
H H
H
j and
[A, A]Ae
?
ρr ρe
H H
H H
H H j
A[A, A]A ιr -[A, A]A ρ -[A, A]
g
*
A
6
g ιR
H H
H H
H H
H j
(15)
wheref andgare the isomorphisms from the above Lemma withX=Y=A, so that the lower right triangles in the above diagrams commute. The morphismsιL and ιR are defined so that the left triangles commute, and note that the upper right
triangles commute, too. Observe that from the definitions of the morphismsf and
g from Lemma3.6we obtainev(A⊗f) =∇Aandev(g⊗A) =∇A.
Similarly as in Corollary 3.4, one proves that f and λ : A[A, A] −→[A, A] are
algebra morphisms so that A[A, A] is a subalgebra of [A, A]. Being both λ and f =λf algebra morphisms andλbeing a monomorphism, one obtains thatf is an isomorphism of algebras. Similarly, so is g. As forλ, one gets that ιl : A[A, A]A −→A[A, A] and λl :Ae[A, A] −→A[A, A] are algebra embeddings, hence so are ιL
andλe. The same holds forιr, ιR andρr, ρe.
In symmetric categories, like for instance in the category of modules over a commutative ring, one works only with one center, without distinguishing its left and right counterparts. The reason is the following.
Proposition 3.7. Let A andB be algebras in a symmetric categoryC and let M andN beA-B-bimodules. The following objects (if they exist) are isomorphic:
A[M, N]B ∼=A⊗B[M, N]∼= [M, N]A⊗B.
Proof. TakeX ∈ C. We consider that the leftA- andA⊗B-module structures and the rightB- andA⊗B-module structures onM⊗X are inherited from those on
M via (2). Recall from2.1that we have left and rightC-isomorphisms of categories
ACB∼=A⊗BC ∼=CA⊗B sinceC is symmetric. This is why the sets in the first row of ACB(M ⊗X, N)∼=A⊗BC(M⊗X, N)∼=CA⊗B(M⊗X, N)
∼
= ∼= ∼=
C(X,A[M, N]B) C(X,A⊗B[M, N]) C(X,[M, N]A⊗B)
are isomorphic. The vertical isomorphisms come from the corresponding adjunc-tions. The claim now follows by uniqueness of right adjoint functors.
4. Morita theory
Morita context as well as the assumptions in Morita theorems. We also study faithfully projective objects making difference between left and right version of the notion and analyze how these are related. In this sectionCwill denote a monoidal category (with unitI) unless otherwise specified.
4.1. Associativity of the tensor product of modules over an algebra. In the literature there have been discrepancies about necessary conditions for the associativity of the (co)tensor product over a (co)algebra. The issue for cotensor products in C was studied in the author’s Ph.D. thesis. Even though the tensor product over an algebra is a dual concept, it gives rise to conditions which are not dual to those obtained in the cotensor product case. For the sake of Morita theorems we give here a complete proof of the associativity of the tensor product over an algebra inC. We will first need:
Definition 4.1. Let A and B be algebras in C. An object M ∈BCA is called A
-coflatif for all algebrasR, S ∈ Cand objectsL∈ACRthe coequalizerM⊗AL∈BCR exists and if the natural morphismM⊗A(L⊗P)−→(M⊗AL)⊗P inBCS, induced by the associativity of the tensor product, is an isomorphism for every P ∈ CS. Symmetrically we define that M is B-coflat. If it is both B- andA-coflat, we say that it isbicoflat.
We have that left adjoint functors preserve coequalizers, [5]. In particular, in a closed category the tensor functor preserves coequalizers.
Remark 4.2. IfC is a category with coequalizers and it is left (right) closed, then anyB-A-bimoduleM isB-coflat (A-coflat). Indeed, for anyX∈RC, L∈ CB, N∈AC and Y ∈ CT it is X ⊗(L⊗B M) ∼= (X ⊗L)⊗B M (resp. (M ⊗A N)⊗Y ∼= M⊗A(N⊗Y)) in C, becauseX⊗ −(resp. − ⊗Y) preserves coequalizers. These isomorphisms are shown to be inRCAandBCT, respectively. Clearly, ifCis closed, thenM is bicoflat. Recall that a braided category is closed if and only if it closed from at least one side.
Lemma 4.3. For every M ∈ CA, N ∈ ACB and L ∈ BC the coequalizers M ⊗A
(N⊗BL)and(M⊗AN)⊗BLare isomorphic provided that the isomorphisms:
X⊗(N⊗BL)= (∼ X⊗N)⊗BL; (16) (M⊗AN)⊗X∼=M⊗A(N⊗X) (17)
hold for anyX ∈ C. These conditions are fulfilled if any of the following holds:
(i) M isA-coflat andL isB-coflat;
(ii) M isA-coflat andC is left closed;
(iii) LisB-coflat andC is right closed;
(iv) C is braided andM isA-coflat andN isB-coflat;
(v) C is braided andL isB-coflat andN isA-coflat;
(vi) C is closed.
Proof. We viewN⊗BLas a leftA-module andM ⊗AN as a rightB-module via
(M⊗AN)⊗BL (M⊗AN)⊗L
ΠM⊗AN,L
µM⊗AN⊗L
(M⊗AN)⊗B⊗L
(M⊗AN)⊗νL
(M⊗N)⊗BL (M⊗N)⊗L
ΠM⊗N,L µM⊗N⊗L
(M⊗N)⊗B⊗L
(M⊗N)⊗νL
6
ΠM,N⊗BL
6
ΠM,N⊗L
6
ΠM,N⊗B⊗L
(M⊗A⊗N)⊗BL (M⊗A⊗N)⊗L
ΠM⊗A⊗N,L µM⊗A⊗N⊗L
(M⊗A⊗N)⊗B⊗L
(M⊗A⊗N)⊗νL
6
(µM⊗N)⊗BL
6
(M⊗νN)⊗BL
66
(µM⊗N)⊗L (M⊗νN)⊗L
6
(µM⊗N)⊗B⊗L
6
(M⊗νN)⊗B⊗L
the three rows are coequalizers. By (17) the second and the third column are coequalizers. Note that all inner rectangles commute. Then by the three by three Lemma we obtain that the first column is a coequalizer too.
In order to show that (M⊗A(N⊗BL),ΠM,N⊗BL) and ((M⊗AN)⊗BL,ΠM,N⊗B
L) are isomorphic as coequalizers, observe the diagram
M⊗A(N⊗BL) M⊗(N⊗BL)
ΠM,N⊗BL
µM⊗(N⊗BL)
M⊗A⊗(N⊗BL)
M⊗νN⊗BL
(M⊗AN)⊗BL (M⊗N)⊗BL
ΠM,N⊗BL (µM⊗N)⊗BL
(M⊗A⊗N)⊗BL,
(M⊗νN)⊗BL
6
χ 6βM,N,L
6
βM⊗A,N,L
whereβ’s are the isomorphisms from (16). The right rectangle commutes obviously with upper lines, but also with lower ones, becauseνN⊗BLis induced byνN⊗BL.
Knowing that both rows are coequalizers, we get that ΠM,N⊗BLβM,N,L inducesχ
so that the left rectangle commutes. Similarly, (ΠM,N ⊗BL)βM,N,L−1 induces the
inverse ofχ.
Note that for the second part of the Lemma by Remark4.2it suffices to discuss the conditions (i), (iv) and (v). Assume we are in the conditions of (iv), let Φ be the braiding ofC. Consider the diagram
N⊗B(L⊗X) N⊗L⊗X
ΠN,L⊗X µN⊗(L⊗X)
N⊗B⊗L⊗X
N⊗(νL⊗X)
(X⊗N)⊗BL X⊗N⊗L
ΠX⊗N,L (X⊗µN)⊗L
X⊗N⊗B⊗L.
(X⊗N)⊗νL
?
θ
?
ΦN⊗L,X
?
ΦN⊗B⊗L,X
The right rectangle commutes obviously both with upper and lower lines. Knowing that both rows are coequalizers, we get that ΠX⊗N,LΦN⊗L,Xinducesθso that the
left rectangle commutes. Similarly, ΠN,L⊗XΦ−N1⊗L,Xinduces the inverse ofθ. Now,
sinceN isB-coflat, we have (X⊗N)⊗BL∼=N⊗B(L⊗X)∼= (N⊗BL)⊗X ∼= X⊗(N⊗BL), applying ΦX,N−1 ⊗BL. Moreover,A-coflatness ofM implies (17), and
we are done with the condition (iv). One can argue symmetrically for (v), and (i)
4.2. Morita theorems. Morita theorems for monoidal categories were developed by Pareigis in [9]. We define here the notion of Morita context that does not appear explicitly in [9].
Definition 4.4.Let A, B∈ C be algebras,P ∈ACB andQ∈BCA. Assume there exist isomorphisms:
(P⊗BQ)⊗AP=∼P⊗B(Q⊗AP) and (Q⊗AP)⊗BQ∼=Q⊗A(P⊗BQ). (18) If the morphismsf :P⊗BQ−→A inACAandg:Q⊗AP −→B inBCB are given so that the diagrams commute, we call the sextuple (A, B, P, Q, f, g) a Morita
(P⊗BQ)⊗AP f⊗AP
-∼
=
A⊗AP
P? ∼
=
P⊗B(Q⊗AP)
P⊗BB ?
P⊗Bg
-∼
=
(19)
(Q⊗AP)⊗BQ g⊗BQ
-∼
=
B⊗BQ
Q? ∼
=
Q⊗A(P⊗BQ)
Q⊗AA ?
Q⊗Af
-∼
=
(20)
contextinC.
If moreover there exist morphisms ζ ∈ C(I, P ⊗BQ) and ξ∈ C(I, Q⊗AP) so that f ◦ζ =idC(I,A) =ηA and g◦ξ = idC(I,B) =ηB, we say that the context is
strict.
Remark 4.5. The isomorphisms in (18) hold if either of the conditions:
(i) P andQare bicoflat;
(ii) P isB-coflat,QisA-coflat andC is left closed;
(iii) P isA-coflat,QisB-coflat andC is right closed;
(iv) P isB-coflat,QisA-coflat andC is braided;
(v) P isA-coflat,QisB-coflat andC is braided;
(vi) C is closed; is fulfilled.
Given a strict Morita context (A, B, P, Q, f, g), from the Morita theorems one can conclude thatP andQare faithfully projective, as we will see further below. Faithfully projective objects can be considered inACor inCBfor algebrasA, B∈ C.
For the sake of completeness, we give here the definitions of both notions which slightly differ from those in [9,11].
Definition 4.6. An object P ∈ AC is called faithfully projective, if A[P, A] and A[P, P] exist, P ∈ ACA[P,P] and A[P, A] ∈ A[P,P]CA are bicoflat, the dual basis morphism db : A[P, A]⊗A P −→ A[P, P] defined via the universal property of
(A[P, P],evP,P)by:
H H
H H
HHj ev⊗AP
?
P⊗db
P⊗A[P, A]⊗AP
P⊗A[P, P] -P∼=A⊗AP
evP,P
and the canonical morphismevˆ :P⊗A[P,P]A[P, A]−→Ainduced by ev :P⊗A[P, A] −→Aare isomorphisms.
Definition 4.7. An object P ∈ CB is called faithfully projective, if [P, B]B and
[P, P]B exist, P ∈ [P,P]BCB and [P, B]B ∈ BC[P,P]B are bicoflat, the dual basis morphism db : P ⊗B [P,B]B −→ [P,P]B defined via the universal property of
([P, P]B,evP,P)by:
H H
H H
H H j
P⊗Bev
? db⊗P
P⊗B[P, B]B⊗P
[P, P]B⊗P -P ∼=P⊗BB
evP,P
(22)
and the canonical morphism ev˜ : [P, B]B⊗[P,P]BP −→B induced by the morphism ev : [P,B]B⊗P −→B are isomorphisms.
Faithfully projective objects inAC and inCB give rise to strict Morita contexts,
as we will see later. In order for this to work, it is evident from Remark 4.5that for faithfully projective objects we need to require complicated conditions (if C
is not closed). To avoid this we decided to include bicoflatness in the above two definitions.
In the following we unify [9, Theorems 5.1 and 5.3] into an equivalence claim including the original results and their right hand-side counterparts. The assump-tions that we find necessary slightly differ from the original ones.
Theorem 4.8. Let C be a monoidal category.
(i) Assume that C is left closed (or braided). The functors F : AC −→ BC and G : BC −→ AC are inverse C-equivalences if and only if there exists a B-coflat object P ∈ ACB and an A-coflat object Q ∈ BCA such that F ∼= Q⊗A− and G ∼= P ⊗B− and (A, B, P, Q, f, g) is a strict Morita context, wheref :P⊗BQ−→Ais an isomorphism inACA andg:Q⊗AP −→B is an isomorphism in BCB.
(ii) Assume that C is right closed (or braided). The functors F : CA −→ CB and G : CB −→ CA are inverse C-equivalences if and only if there exists an A-coflat object P ∈ ACB and a B-coflat object Q ∈ BCA such that F ∼= − ⊗AP and G ∼= − ⊗BQ and (A, B, P, Q, f, g) is a strict Morita context, wheref :P⊗BQ−→Ais an isomorphism inACA andg:Q⊗AP −→B is an isomorphism in BCB.
(iii) Assume that C is closed. The two pairs of functors F : AC −→ BC and G : BC −→ AC; and F0 : CA −→ CB and G0 : CB −→ CA, are inverse C-equivalences if and only if there exist objects P ∈ACB and Q∈BCA such that F ∼=Q⊗A−andG ∼=P⊗B−, andF0∼=− ⊗AP andG0∼=− ⊗BQ, and(A, B, P, Q, f, g)is a strict Morita context, wheref :P⊗BQ−→Ais an isomorphism in ACA andg:Q⊗AP −→B is an isomorphism in BCB.
Namely, in the case of (i), for anyX ∈BCwe haveQ⊗A(P⊗BX)∼= (Q⊗AP)⊗BX
sinceQis A-coflat andC is left closed, or sinceQisA-coflat,P isB-coflat and C
is braided, due to Lemma4.3. Similarly forP⊗B(Q⊗AY)∼= (P⊗BQ)⊗AY for
anyY ∈AC. The reasoning for (ii) is similar.
Theorem 4.9. [11, Theorem 2.1]Let (A, B, P, Q, f, g) be a strict Morita context inC and assume thatP andQare bicoflat. Then:
(i) There are isomorphismsP∼= [Q, A]A∼=B[Q, B]inACB andQ∼= [P, B]B ∼= A[P, A] inBCA.
(ii) There are isomorphismsA∼= [P, P]B∼=B[Q, Q]inACAandB∼= [Q, Q]A∼= A[P, P]in BCB that are also isomorphisms of algebras.
(iii) AP, PB, BQandQA are faithfully projective.
Observe that in the proof of (iii) above, the diagrams (19) and (20) translate into the diagram (21) and the one determining morphism ˆev :P⊗A[P,P]A[P, A]−→A
(and (22) and the diagram determining ˜ev : [P, B]B ⊗[P,P]BP −→B). Note that
by assumptionP andQare bicoflat, as in our definition of faithful projectiveness. Translating the diagrams in the reversed order, one proves:
Proposition 4.10. Let C be a monoidal category.
(i) IfPB is faithfully projective, then ([P, P]B, B, P,[P, B]B,db,ev˜)is a strict Morita context, with db andev from Definition˜ 4.7.
(ii) IfAP is faithfully projective, then(A,A[P, P], P,A[P, A],evˆ,db)is a strict Morita context, withev and db from Definitionˆ 4.6.
From here and Theorem4.8we have:
Corollary 4.11. (i) If PB is faithfully projective and C is left closed (or braided), then the functorP⊗B−:BC −→[P,P]BC is aC-equivalence.
(ii) If AP is faithfully projective and C is right closed (or braided), then the functor− ⊗AP :CA−→ CA[P,P] is aC-equivalence.
Let us now study faithfully projective objects withA=B=I. In order to distin-guish two kinds of faithful projectiveness we will write{M, N}and [M, N] for the two inner-hom objects (recall2.2).
An object faithfully projective inICwe callleft faithfully projectiveand an object
faithfully projective in CI we call right faithfully projective. Note that an object P is right faithfully projective if P∈ [P,P]C and [P, I]∈ C[P,P] are coflat, the dual
basis morphismdb:P⊗[P,I]−→[P,P] defined viaevP,P(db⊗P)=P⊗evP,I is an
isomorphism (this defines the concept ofright finite object) and ˜ev:[P, I]⊗[P,P]IP −→Iinduced byev:[P,I]⊗P −→I is an isomorphism. We have a similar statement for a left faithfully projective object, and a left finite object is defined similarly as a right one.
With A=B=I the requirement made in Corollary 4.11, that C be left (right) closed or braided, becomes superfluous in view of the comment after Theorem4.8. Thus we may write:
Corollary 4.13. (i) If P is a right faithfully projective object in a monoidal categoryC, then([P, P], I, P,[P, I],db,ev˜)is a strict Morita context. More-over, the functor P⊗ −:C −→[P,P]C is aC-equivalence.
(ii) If P is a left faithfully projective object in a monoidal category C, then
(I,{P, P}, P,{P, I},evˆ,db)is a strict Morita context. Moreover, the func-tor− ⊗P :C −→ C{P,P} is aC-equivalence.
For the rest of this subsection assume that C is a braided monoidal category with braiding Φ. In the above CorollaryP is a left [P, P]-module byev: [P, P]⊗P −→P andP∗= [P, I] is a right [P, P]-module by
[P,I] [P,P] P
=
[P,I] [P,P] P
h
ev
(23)
whereasP is a right {P, P}-module byev :P⊗ {P, P} −→P, andP∗ ={P, I} is a left{P, P}-module by
P {P,P} {P,I}
PP
h
ev
=
P {P,P} {P,I}
h
ev
h
ev
which is
P {P,P}P∗ PP
=
P [P,P]{P,I}
ev
Composing the latter with Φ−P,1{P,P}⊗P∗ and then Φ
−1
{P,P},P∗⊗P from above, we
obtain:
{P,P}P∗ P
PP =
[P,P]P∗P
ev nat.=
[P,P]P∗P
ev and
P∗{P,P}P
PP
=
P∗[P,P] P
ev
(23)
=
P∗ [P,P] P
respectively. By the universal property of ([P, P], ev) this yields:
P∗{P,P}
PP
P∗
= P∗ [P,P]
P∗
which implies
{P,P}P∗
PP
P∗
=
[P,P]P∗
P∗
(24)
Now consider the diagram
-µP∗⊗P
P∗⊗[P, P]⊗P
-P∗⊗νP
? Φ−P,P1∗
P∗⊗P P∗⊗
[P,P]P
-Π1
P⊗{P,P?}P∗ θ
P⊗P∗
-Π2
6
ψ
-µP⊗P∗
P⊗ {P, P} ⊗P∗
-P⊗νP∗
ev
H H
HHj
I ev * ˜ ev H H H H Y ev˜ 1
2
3
We want to define morphisms θ and ψ so that the right rectangle in the above picture commutes. The two rows are coequalizers. From the second one we have:
P{P,P}P∗ Π2
P⊗{P,P}P∗ =
P{P,P}P∗ PP Π2
P⊗{P,P}P∗
which by (24) is equivalent to
P [P,P]P∗
ev
Π2
P⊗{P,P}P∗
=
P [P,P]P∗
Π2
P⊗{P,P}P∗
Composing to this identity Φ−P,1[P,P] ⊗P∗ and then Φ−[P,P1 ]⊗P,P∗ from above, we
obtain:
[P,P] P P∗
h
ev
Π2
P⊗{P,P}P∗
=
[P,P] P P∗
Π2
P⊗{P,P}P∗
and
P∗[P,P]P
ev
Π2
P⊗{P,P}P∗
= P∗[P,P]P
Π2
P⊗{P,P}P∗
respectively. The latter is equivalent to
P∗[P,P]P
PP
Π2
P⊗{P,P}P∗
= P∗[P,P]P
Π2
P⊗{P,P}P∗
= P∗[P,P]P
Π2
P⊗{P,P}P∗
This proves that Π2Φ−1 induces θ so thatθΠ1 = Π2Φ−1. On the other hand, by
the first coequalizer in (25) we have:
P∗[P,P]P
Π1
P∗⊗ [P,P]P
=
P∗ [P,P] P
h
ev
Π1
P∗⊗ [P,P]P
and this implies
P [P,P]P∗
Π1
P∗⊗ [P,P]P
=
P [P,P]P∗
ev
Π1
P∗⊗ [P,P]P
(26)
Then we find:
P{P,P}P∗
Π1
P∗⊗ [P,P]P
=
P{P,P}P∗
ev
Π1
P∗⊗ [P,P]P
=
P [P,P]P∗
ev
Π1
P∗⊗ [P,P]P
(26)
=
P [P,P]P∗
Π1
P∗⊗ [P,P]P
=
P [P,P]P∗
Π1
P∗⊗ [P,P]P
(24)
=
P{P,P}P∗ PP
Π1
P∗⊗ [P,P]P
This proves that Π1Φ induces ψ so thatψΠ2 = Π1Φ. Now, ψθΠ1 =ψΠ2Φ−1 =
Π1ΦΦ−1 = Π1, which yields ψθ =idsince Π1 is an epimorphism. Similarly, one
provesθψ=id.
From the commutativity of the inner diagramsh1i,h2iandh3iin (25) we obtain ˜
evθΠ1= ˜evΠ2Φ−1 = ˜evΠ1, hence ˜evθ= ˜ev. Sinceθ is an isomorphism, we have
that ˜ev is an isomorphism if and only if ˜ev is an isomorphism. We have proved:
4.3. The central part of the Morita theorems. In [9] Pareigis proved that for two algebras A and B in a symmetric category C for which the categories of modules AC and BC are C-equivalent the centers A[A, A]A and B[B, B]B are
isomorphic as commutative algebras. This is proved using a more general result. LetU,V:AC −→BCbe twoC-functors such thatU ∼=P⊗A−andV ∼=Q⊗A−are C-isomorphisms for some B-A-bimodules P and Q. Then PA and QA are coflat.
ForY ∈ C the functorU ⊗Y :AC −→BC is defined by (U ⊗Y)(M) :=U(M ⊗Y)
forM ∈AC. It is aC-functor. Furthermore, one defines a (contravariant) functor
[U,V] :C −→ Set by [U,V](Y) :=C-N at(U ⊗Y,V), the latter denoting the family ofC-natural transformationsU ⊗Y −→ V, and forh∈ C(Z, Y) one defines [U,V](h) by [U,V](h)(ϕ)(M) :=ϕ(M)◦ U(M⊗h) whereϕ∈ C-N at(U ⊗Y,V).
Proposition 4.15. [9, Theorem 6.1] In a symmetric monoidal category C there is a natural isomorphism of functors [U,V]∼=BCA(P⊗ −, Q). If B[P, Q]A exists, then[U,V](Y)∼=C(Y,B[P, Q]A).
The Proposition remains valid also when C is a braided category. In its proof the braiding is not used, however in order to assure thatU ⊗Y is aC-functor in the non-symmetric case, we need to keep track on the braiding. Consider the diagram:
-µP⊗M⊗Y
P⊗A⊗M⊗Y
-P⊗νM⊗Y
?
P⊗ΦM,Y
P⊗M⊗Y -P⊗A(M⊗Y) ΠP,M⊗Y
?
β
P⊗Y ⊗M -(P⊗Y)⊗AM. ΠP⊗Y,M
?
P⊗ΦA⊗M,Y
-µP⊗Y ⊗M
P⊗Y ⊗A⊗M
-P⊗Y ⊗νM
The left rectangle commutes with lower lines by naturality, and also with upper lines, given that the rightA-module structure ofP⊗Y is given as in (2). Since both horizontal rows are coequalizers, we obtain the isomorphismβ. Then (U ⊗Y)(M) =
U(M ⊗Y) = P ⊗A(M ⊗Y) ∼= (P ⊗Y)⊗AM and hence (U ⊗Y)(M ⊗X) ∼=
(P⊗Y)⊗A(M⊗X)∼= ((P⊗Y)⊗AM)⊗X∼= (U ⊗Y)(M)⊗X. The penultimate
isomorphism is a consequence of the isomorphismβ and of the A-coflatness ofP. This proves thatU ⊗Y is aC-functor also whenC is braided.
Applying Proposition4.15to the identity functorU=V= IdA, one obtains
[AId,AId]∼=ACA(A⊗ −, A). (27)
Suppose that F :AC −→ BC is a C-equivalence. In [9, Corollary 6.3] it is proved
that [AId,AId](Y) and [BId,BId](Y) are isomorphic for Y ∈ C. In the proof the
braiding is used only once at a computation in order to intertwine two objects
X, Y ∈ C when proving the C-functor property, but the sign of the braiding is irrelevant. Now, ifA[A, A]AandB[B, B]B exist, they both are representing objects
for the same functor, due to (27), so they are isomorphic. Just before [9, Corollary 6.3] the multiplication and unit for the functors in (27) is described and it is concluded that this is an isomorphism of “monoids”. On the objectA[A, A]Athese
Observe that in the above construction we have worked with right C-functors. In2.1we saw that the category BCA is isomorphic as a rightC-category toB⊗AC.
Then the isomorphism in Proposition 4.15can be rewritten as follows: [U,V] ∼=
B⊗AC(P⊗−, Q), and the one in (27) as [AId,AId]∼=A⊗AC(A⊗−, A). The result of
the above paragraph can then be restated as: A⊗A[A, A]∼=B⊗B[B, B] as algebras. By symmetry, starting with the functors U0 ∼= − ⊗B P, V0 ∼= − ⊗BQ and
(Y ⊗ U0)(M) := U0(Y ⊗M) for M ∈ CB, one may prove the isomorphisms of
functors [U0,V0]∼=C
B⊗A(− ⊗P, Q) and [IdB,IdB]∼=CB⊗B(− ⊗B, B). Given a C
-equivalence betweenCAandCB, the latter yields, analogously as above, the algebra
isomorphism [A, A]A⊗A∼= [B, B]B⊗B. We now may claim:
Theorem 4.16. Let C be a braided monoidal category andA, B∈ C algebras.
(i) If AC and BC are C-equivalent, then the left centers of A and B (if they exist) are isomorphic as commutative algebras.
(ii) If CA andCB are C-equivalent, then the right centers ofA and B (if they exist) are isomorphic as commutative algebras.
5. Azumaya algebras
In [2] we presented the construction of the Brauer group of a braided monoidal categoryCusing the notation of braided diagrams. Azumaya algebras inC, whose equivalence classes form the Brauer group ofC, were characterized in [11, Theorem 3.1]. However, an explicit proof is omitted and it is not stated which kind of faithful projectiveness is used. Observe that faithful projectiveness ofP inCB can not be
deduced from the one in AC, and vice versa. Although in Proposition 4.14 we
proved that IP is faithfully projective if and only if so is PI, given that to the
author’s knowledge this was not discussed in the literature so far, we present below the full proof of a reformulated [11, Theorem 3.1] studying how both left and right faithful projectiveness of the (Azumaya) algebra in question are involved.
Theorem 5.1. LetC be a closed braided monoidal category andA∈ Can algebra. The following are equivalent:
(i) The functors A⊗ − : C −→ A⊗AC and − ⊗A : C −→ CA⊗A establish C-equivalences of categories;
(ii) MorphismsF:A⊗A−→[A, A] andG:A⊗A−→[A, A] defined via:
A⊗A A
F
h
ev
A =
A A A
A
(28) and
A A⊗A
G
h
ev
A =
A A A
A
(29)
are isomorphisms andA is left and right faithfully projective.
Proof. From the C-equivalence A⊗ −: C −→A⊗AC, by Theorem4.8 part (i) and Theorem4.9part (i) we have a strict Morita context (I, A⊗A, A∗, A,ev˜, db). By part (iii) it follows thatAI is faithfully projective, i.e. Ais right faithfully
Theorem4.9part (i), yields a strict Morita context (I, A⊗A, A, A∗,ev, db). From here we get thatIAis faithfully projective, i.e. A is left faithfully projective.
The algebra isomorphismω :B −→[Q, Q]A from Theorem4.9part (ii) is given by idQ=:ev(ω⊗B Q) : (Q∼=B⊗BQ
ω⊗BQ
−→ [Q,Q]A⊗BQ ev
−→ Q). The latter is induced as a morphism on the coequalizer by the commuting diagram:
-∇B⊗Q
B⊗B⊗Q
-B⊗ν
?
ω⊗Q
B⊗Q -B⊗BQ∼=Q
ΠB,Q
ν
P PP
PP PP
PP P
PPq
Q?
idQ
[Q, Q]A⊗Q ev
-Observe thatωis defined in terms of the actionν ofBonQ, whereas on the other hand ν induces idQ. Substituting in our situation using the first Morita context B=A⊗A, Q=A and A=I, we obtain that the algebra isomorphism F: A⊗A
−→[A, A] is induced by the left actionµlofA⊗AonA, that isev(F⊗A) =µl, which
by (3) yields (28). Similarly, consider the algebra isomorphism θ : B −→ A[P, P] from Theorem4.9part (ii) defined viaev(P⊗Bθ) =idP and substitute from our
second Morita context B = A⊗A, P = A and A = I. Recalling from2.5 that
I[A, A] = {A, A} ∼= [A, A] as algebras, we obtain that the algebra isomorphism G:A⊗A−→[A, A] is induced by the right action µrof A⊗AonA. This means ev(A⊗G) =µr, and by (4) we get (29).
Conversely, left and right faithful projectiveness of A, by Corollary 4.13imply respectively that − ⊗A : C −→ C[A,A] and A⊗ −: C −→[A,A]C are C-equivalences.
Using the algebra isomorphisms G:A⊗A −→[A, A] and F:A⊗A −→[A, A], we
obtain the claim.
An algebra satisfying one of the conditions from the above Theorem is called an
Azumaya algebra. The right inverse functors forA⊗ −and− ⊗Aare the functors
Ae[A,−]∼= (−)Aand [A,−]Ae ∼=A(−), respectively.
The morphismsF :A⊗A−→[A, A] andG:A⊗A−→[A, A] are in general not equal. However, if the category is symmetric, they coincide:
A⊗A A
G
h
ev
A =
A⊗A A
G
h
ev
A
nat.(29)
=
A A A
A
nat.
=
A A A
A
ass.
=
A A A
A
which since C is symmetric equals (28). By the universal property of ([A, A], ev) it follows F =G. The structures chosen in (3) and (4) (ΦB,M in (3) and ΦM,A
in (4)) assure that F and G are morphisms of algebras in any braided monoidal category.
A[A,−]A. In Proposition3.7we showed that in a symmetric category all the three
equivalence functors are mutually equivalent.
We close this section by proving that an Azumaya algebra is central.
Definition 5.2. An algebraA∈ C is calledleft central if its left centerA⊗A[A, A]
is trivial, that is, isomorphic toI. Similarly, one defines a right central algebra. A is calledcentral if it is both left and right central.
The following claim is implicitly contained in [11, Theorem 4.9, (3)⇒(1)], where the category was assumed to be abelian. Our proof illustrates how right faithful projectiveness ofAimplies thatAis left central, whereas left faithful projectiveness ofAimplies that it is right central.
Proposition 5.3. An Azumaya algebra A in a braided monoidal category C is central.
Proof. From the right faithful projectiveness ofAwe have the strict Morita context ([A, A], I, A,[A, I],db,ev˜). Applying Theorem4.9part (ii), we getI∼=[A,A][A, A].
Taking into account the algebra isomorphism F : A⊗A −→ [A, A], we obtain
I ∼= A⊗A[A, A]. On the other hand, the left faithful projectiveness of A yields the strict Morita context (I,{A, A}, A,{A, I},evˆ,db). From Theorem4.9part (ii) we get I ∼= [A, A][A,A]. This, together with the algebra isomorphism G : A⊗A
−→[A, A], impliesI∼= [A, A]A⊗A.
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B. Femi´c
Facultad de Ciencias 11 400 Montevideo, Uruguay
