Theory and Applications of Categories, Vol. 22, No. 4, 2009, pp. 77–96.
ON ENDOMORPHISM ALGEBRAS OF SEPARABLE MONOIDAL
FUNCTORS
BRIAN DAY AND CRAIG PASTRO
Abstract. We show that the (co)endomorphism algebra of a sufficiently separable
“fibre” functor into Vectk , for k a field of characteristic 0, has the structure of what
we call a “unital” von Neumann core in Vectk . For Vectk , this particular notion of
algebra is weaker than that of a Hopf algebra, although the corresponding concept in
Set is again that of a group.
1. Introduction
Let C = (C , ⊗, I, c) be a braided (or even symmetric) monoidal category. Recall that
an algebra in C is an object A ∈ C equipped with a multiplication µ : A ⊗ A → A and
a unit η : I → A satisfying µ3 = µ(1 ⊗ µ) = µ(µ ⊗ 1) : A⊗3 → A (associativity) and
µ(η ⊗ 1) = 1 = µ(1 ⊗ η) : A → A (unit conditions). Dually, a coalgebra in C is an object
C ∈ C equipped with a comultiplication δ : C → C ⊗ C and a counit : C → I satisfying
δ3 = (1 ⊗ δ)δ = (δ ⊗ 1)δ : C → C ⊗3 (coassociativity) and ( ⊗ 1)δ = 1 = (1 ⊗ )δ : C → C
(counit conditions).
A very weak bialgebra in C is an object A ∈ C with both the structure of an algebra
and a coalgebra in C related by the axiom
δµ = (µ ⊗ µ)(1 ⊗ c ⊗ 1)(δ ⊗ δ) : A ⊗ A → A ⊗ A.
For example, when C = Vectk , any k-bialgebra or weak k-bialgebra is a very weak
bialgebra in this sense.
We note briefly that, if A is such a structure, but has no unit or counit, we simply
call A a semibialgebra, or core for short. This minimal structure on A is then called a
von Neumann core in C if it also is equipped with an endomorphism S : A → A in C
satisfying the axiom
µ3 (1 ⊗ S ⊗ 1)δ3 = 1 : A → A.
The first author gratefully acknowledges partial support of an Australian Research Council grant,
while the second author is partially supported by GCOE, Kyoto University. Parts of this work were
completed while the second author was a Ph.D. student at Macquarie University, Australia, and during
that time was gratefully supported by an international Macquarie University Research Scholarship and
a Scott Russell Johnson Memorial Scholarship.
The authors would like to thank Ross Street for several helpful comments.
Received by the editors 2007-12-21 and, in revised form, 2009-04-17.
Transmitted by Jean-Louis Loday. Published on 2009-04-20.
2000 Mathematics Subject Classification: 18D99, 16B50.
Key words and phrases: separable fibre functor, Tannaka reconstruction, bialgebra, von Neumann
core.
c Brian Day and Craig Pastro, 2009. Permission to copy for private use granted.
77
78
BRIAN DAY AND CRAIG PASTRO
A von Neumann regular semigroup is precisely a von Neumann core in Set, while the free
k-vector space on it is a special type of von Neumann core in Vectk . However, within
this article we shall always suppose that A has both a unit and a counit. For example,
when C = Vectk , a Hopf k-algebra or a weak Hopf k-algebra is a von Neumann core in
this somewhat stronger sense.
Since groups A in Set are characterized by the (stronger) axiom
1 ⊗ η = (1 ⊗ µ)(1 ⊗ S ⊗ 1)δ3 : A → A ⊗ A,
(†)
a very weak bialgebra A satisfying (†), in the general C , will be called a unital von
Neumann core in C . Such a unital von Neumann core A always has a left inverse to the
“fusion” operator [9]
(1 ⊗ µ)(δ ⊗ 1) : A ⊗ A → A ⊗ A,
namely
(1 ⊗ µ)(1 ⊗ S ⊗ 1)(δ ⊗ 1) : A ⊗ A → A ⊗ A.
Any Hopf algebra in C satisfies the stronger axiom (†), but a weak Hopf algebra does
not necessarily do so. In this article we are mainly interested in producing a unital von
Neumann core, namely End∨ U , associated to a certain type of split monoidal functor
U into Vectk . It seems unlikely that all unital von Neumann cores in Vectk may be
reproduced as such.
We will tacitly assume throughout the article that the ground category [8] is Vect =
Vectk , for k a field of characteristic 0, so that the categories and functors considered
here are all k-linear (although any reasonable category [D, Vect] of parameterized vector
spaces would suffice). We denote by Vectf the full subcategory of Vect consisting of the
finite dimensional vector spaces, and we further suppose that C = (C , ⊗, I, c) is a braided
monoidal category with a “fibre” functor
U : C → Vect,
with both a monoidal structure (U, r, r0 ) and a comonoidal structure (U, i, i0 ), which need
not be inverse to one another. We call U separable 1 if ri = 1 and i0 r0 = dim(U I) · 1; i.e.,
for all A, B ∈ C , the diagrams
i
/ UA ⊗ UB
OOO
OOO
OOO
r
OOO
1
'
U (A ⊗ B)
U (A ⊗ B)
1
r0
/ UI
GG
GG
GG
i0
dim U I·1 GGG
G# k GG
k
Strictly, we should also require the conditions (cf. [1, 10])
(r ⊗ 1)(1 ⊗ i) = ir : U A ⊗ U (B ⊗ C) → U (A ⊗ B) ⊗ U C, and
(1 ⊗ r)(i ⊗ 1) = ir : U (A ⊗ B) ⊗ U C → U A ⊗ U (B ⊗ C)
in order for U to be called “separable”, but we do not need these here.
ON ENDOMORPHISM ALGEBRAS OF SEPARABLE MONOIDAL FUNCTORS
79
commute.
First we produce an algebra structure (µ, η) on
Z C
∨
U (C)∗ ⊗ U C
End U =
using the monoidal and comonoidal structures on U . Secondly, we suppose that C has
a suitable small generating set A of objects, and produce a coalgebra structure (δ, ) on
End∨ U when each value U A, A ∈ A , is finite dimensional. Finally, we assume that each
A ∈ A has a ⊗-dual A∗ which also lies in A , and that U is equipped with an isomorphism
U (A∗ ) ∼
= U (A)∗
for all A ∈ A . This isomorphism should be suitably related to the evaluation and
coevaluation maps of C and Vectf which then allows us to define a natural non-degenerate
form
U (A∗ ) ⊗ U A → k.
This last assumption is sufficient to provide End∨ U with an automorphism S so that it
becomes a unital von Neumann core in the above sense whenever (U, r, r0 ) is a braided
monoidal functor.
By way of examples, we note that many separable monoidal functors are constructable
from separable monoidal categories, i.e., from monoidal categories C for which the tensor
product map
⊗ : C (A, B) ⊗ C (C, D) → C (A ⊗ C, B ⊗ D)
is a naturally split epimorphism (as is the case for some finite cartesian products such as
Vectnf ). A closely related source of examples is the notion of a weak dimension functor
on C (cf. [6]); this is a comonoidal functor
(d, i, i0 ) : C → Setf
for which the comonoidal transformation components
i = iC,D : d(C ⊗ D) → dC × dD
are injective functions, while the unique map i0 : dI → 1 is surjective. Various examples
are described at the conclusion of the paper.
We suppose the reader is familiar to some extent with the standard Tannaka reconstruction problem when restricted to the case of U strong monoidal (see [7] for example).
2. The very weak bialgebra End∨ U
If C is a (k-linear) monoidal category and
U : C → Vect
80
BRIAN DAY AND CRAIG PASTRO
has a monoidal structure (U, r, r0 ) and a comonoidal structure (U, i, i0 ), then End∨ U ,
when it exists, has an associative and unital k-algebra structure whose multiplication µ
is the composite map
Z C
Z D
Z B
µ
∗
∗
/
U (C) ⊗ U C ⊗
U (D) ⊗ U D
U (B)∗ ⊗ U B
O
∼
=
Z
C,D
R⊗
U (C)∗ ⊗ U (D)∗ ⊗ U C ⊗ U D
can
Z
C,D
(U C ⊗ U D) ⊗ U C ⊗ U D
Z
/
∗
R
C,D
i∗ ⊗r
U (C ⊗ D)∗ ⊗ U (C ⊗ D)
while the unit η is given by
k
η
/
Z
C
U (C)∗ ⊗ U C
O
∼
=
coprC=I
k∗ ⊗ k
i∗0 ⊗r0
/
U I ∗ ⊗ U I.
The associativity and unit axioms for (End∨ U, µ, η) now follow directly from the corresponding associativity and unit axioms for (U, r, r0 ) and (U, i, i0 ). An augmentation is
given by
Z C
/k
U (C)∗ ⊗ U C
A
]::
::
:
coprC=D ::
e
U (D)∗ ⊗ U D
in Vect, where e denotes evaluation in Vect.
We also observe that the coend
Z C
∨
End U =
U (C)∗ ⊗ U C
actually exists in Vect if C contains a small full subcategory A with the property that
the family
{U f : U A → U C | f ∈ C (A, C), A ∈ A }
is epimorphic in Vect for each object C ∈ C . In fact, we shall use the stronger condition
that the maps
Z A∈A
αC :
C (A, C) ⊗ U A → U C
ON ENDOMORPHISM ALGEBRAS OF SEPARABLE MONOIDAL FUNCTORS
81
should be isomorphisms,
not justR epimorphisms. This stronger condition implies that we
R C∈C
A∈A
can effectively replace
by
since by the Yoneda lemma
Z C
Z C
Z A
∗
∗
∼
U (C) ⊗ U C =
U (C) ⊗ (
C (A, C) ⊗ U A)
Z A
∼
U (A)∗ ⊗ U A.
=
If we furthermore ask that each value U A be finite dimensional for A in A , then
Z A∈A
∨
∼
End U =
U (A)∗ ⊗ U A
is canonically a k-coalgebra with counit the augmentation , and comultiplication δ given
by
Z A
Z A
Z A
δ /
∗
∗
U (A)∗ ⊗ U A
U (A) ⊗ U A ⊗
U (A) ⊗ U A
O
O
copr
copr ⊗ copr
U (A)∗ ⊗ U A
1⊗n⊗1
/
U (A)∗ ⊗ U A ⊗ U (A)∗ ⊗ U A,
where n denotes the coevaluation morphism in Vectf .
2.1. Proposition. If U is separable then End∨ U satisfies the k-bialgebra axiom expressed by the commutativity of
δ⊗δ
End∨ U ⊗ End∨ U
/
(End∨ U )⊗4
1⊗c⊗1
(End∨ U )⊗4
µ
µ⊗µ
End∨ U
δ
/
End∨ U ⊗ End∨ U.
Proof. Let B denote the monoidal full subcategory of C generated by A (we will
essentially replace C by this small category B). Then, for all C, D in B, we have, by
induction on the tensor lengths of C and D, that U (C ⊗ D) is finite dimensional since it
is a retract of U C ⊗ U D. Moreover, we have
Z A∈A
Z B∈B
∗
∼
U (A) ⊗ U A =
U (B)∗ ⊗ U B
by the Yoneda lemma, since the natural transformation
Z A∈A
α = αB :
C (A, B) ⊗ U A → U B
82
BRIAN DAY AND CRAIG PASTRO
is an isomorphism for all B ∈ B. Since ri = 1, the triangle
7
noooo
(U C ⊗ U D) ⊗ (U C ⊗ U D)∗
o
ooo
OOO
n OOO
'
k OOOO
r⊗i∗
U (C ⊗ D) ⊗ U (C ⊗ D)∗ ,
commutes in Vectf , where n denotes the coevaluation maps. The asserted bialgebra
axiom then holds on End∨ U since it reduces to the following diagram on filling in the
definitions of µ and δ (where, for the moment, we have dropped the symbol “⊗”):
U C U (C)∗ U D U (D)∗
∼
=
1 n 1 1 n 1
U C U D (U C U D)∗
r i∗
U (C D) U (C D)∗
U C (U C U (C)∗ ) U (C)∗ U D (U D U (D)∗ ) U (D)∗
∼
=
U C U D U (C)∗ U (D)∗
∼
=
/
U C U D U C U D U (C)∗ U (D)∗ U (C)∗ U (D)∗
/
1 n 1
U C U D U C U D (U C U D)∗ (U C U D)∗
/
1 n 1
∼
=
r r i ∗ i∗
U (C D) U (C D) U (C D)∗ U (C D)∗
for all C, D ∈ B.
Notably the bialgebra axiom expressed by the commutativity of
µ
∨
/ End∨ U
End∨ U ⊗ End
GG U
GG

GG


GG

⊗ GG
G# 
k
does not hold in general, while the form of the axiom expressed by
End∨ U
aD
δ
DD
DD
D
η DDD
D
k
/
End∨ U9 ⊗ End∨ U
rr
rrr
r
r
rrrη⊗η
rrr
holds only if η = r0 ⊗ i∗0 . Also η = dim U I · 1 for U separable.
The single k-bialgebra axiom established in the above proposition implies that the
“fusion” operator (1 ⊗ µ)(δ ⊗ 1) : A ⊗ A → A ⊗ A satisfies the fusion equation (see [9] for
details).
The k-linear dual of End∨ U is of course
Z
Z
C
∗
∼
U (C) ⊗ U (C), k = [U (C)∗ , U (C)∗ ]
C
ON ENDOMORPHISM ALGEBRAS OF SEPARABLE MONOIDAL FUNCTORS
83
which is the endomorphism k-algebra of the functor
U (−)∗ : C op → Vect.
If ob A is finite, so that
Z
is finite dimensional, then
Z
A
U (A)∗ ⊗ U A
[U (C) , U (C) ] ∼
=
∗
∗
C
Z
[U (A)∗ , U (A)∗ ]
A
is also a k-coalgebra.
3. The unital von Neumann core End∨ U
We now take C = (C , ⊗, I, c) to be a braided monoidal category and A ⊂ C to be a
small full subcategory of C for which the monoidal and comonoidal functor U : C → Vect
induces
U : A → Vectf
on restriction to A . We suppose that A is such that
• the identity I of ⊗ lies in A , and each object of A ∈ A has a ⊗-dual A∗ lying in
A.
With respect to U , we suppose A has the properties
• “U -irreducibility”: A (A, B) 6= 0 implies dim U A = dim U B for all A, B ∈ A ,
• “U -density”: the canonical map
Z A∈A
αC :
C (A, C) ⊗ U A → U C
is an isomorphism for all C ∈ C ,
• “U -trace”: each object of A has a U -trace in C (I, I), where by U -trace of A ∈ A
we mean an isomorphism d(A) in C (I, I) such that the following two diagrams
commute.
I
d(A)
n
/
k
IO
e
A ⊗ A∗
c
/
A∗ ⊗ A
/
dim U A
k
r0
r0
UI
dim U I·U (d(A))
/
UI
We shall assume dim U I 6= 0 so that the latter assumption implies dim U A 6= 0, for
all A ∈ A .
84
BRIAN DAY AND CRAIG PASTRO
We require also a natural isomorphism
∼
=
u = uA : U (A∗ ) −
→ U (A)∗
such that
r0
k
/
UI
n
U A ⊗ U (A)∗
Un
U (A ⊗ A∗ )
AA
AA
AA
AA
−1
1⊗u
>
}}
}
}
}} r
}}
(n, r, r0 )
U A ⊗ U (A∗ )
commutes, and
i0
UO I
/
kO
e
Ue
U (A)∗ ⊗ U A
U (A∗ ⊗ A)
AA
AA
AA
i AA
}>
}}
}
}}
}} u⊗1
(e, i, i0 )
U (A∗ ) ⊗ U A
commutes. This means that U “preserves duals” when restricted to A .
An endomorphism
σ : End∨ U → End∨ U
may be defined by components
Z
A
σ
∗
U (A) ⊗ U A
/
Z
A
U (A)∗ ⊗ U A
O
O
copr
copr
σA
U (A)∗ ⊗ U A
/ U (A∗ )∗
⊗ U (A∗ ),
each σA being given by commutativity of
U (A)∗ ⊗ U A
σA
/ U (A∗ )∗
⊗ U (A∗ )
O
c
1⊗ρ
U (A)∗ ⊗ U (A)∗∗
u−1 ⊗u∗
/ U (A∗ ) ⊗ U (A∗ )∗ ,
where ρ denotes the canonical isomorphism from a finite dimensional vector space to its
double dual. Clearly each component σA is invertible.
ON ENDOMORPHISM ALGEBRAS OF SEPARABLE MONOIDAL FUNCTORS
85
3.1. Theorem. Let C , A , and U be as above, and suppose that U is braided and
separable as a monoidal functor. Then there is an automorphism S on End∨ U such that
(End∨ U, µ, η, δ, , S) is a unital von Neumann core in Vectk .
Proof. A family of maps {SA | A ∈ A } is defined by
SA = dim U I · (dim U A)−1 · σA .
Then, by the U -irreducibility assumption on the category A , this family induces an
automorphism S on the coend
∞ Z A∈An
X
∨
∼
End U =
U (A)∗ ⊗ U A,
n=1
where An is the full subcategory of A determined by {A | dim U A = n}. We now take S
to be the prospective core endomorphism on End∨ U and check that
1 ⊗ η = (1 ⊗ µ)(1 ⊗ S ⊗ 1)δ3 .
From the definition of µ and δ, we require commutativity of the exterior of the following
diagram (where, again, we have dropped the symbol “⊗”):
1 1 SA 1 1
/ U (A)∗ U A U (A∗ )∗ U (A∗ ) U (A)∗ U A
U (A)∗ U A U (A)∗ U AQ U (A)∗ U A
O
h QQ
O
QQQ1 1 c 1 1
QQQ
QQQ
(2)
Q
1 1 u∗ u−1 1 1
∗
∗
∗
U (A) U A U A U (A) U (A) U A
O
(1)
1 n 1 1 1
U (A)∗ U A U (A)∗∗ U (A)∗ U (A)∗ U A
1 1 1 c 1
O
U (A)∗ U A U6 A U (A)∗ U (A)∗ U A
mm
mmm
m
m
mmm
mmm 1 1 n 1 1
/
U (A)∗ U A U (A)∗ U A
O
1 1 1 c 1
U (A)∗ U A U (A)∗∗ U (A)∗ U (A)∗ U A
dim U I·(dim U A)−1 ·(1 1 e∗ 1 1)
U (A)∗ U A U (A∗ )∗ U (A∗ ) U (A)∗ U A
1 n 1
∼
=
U (A)O∗ U A
U (A)∗ U A (U (A∗ ) U A)∗ U (A∗ ) U A
(3)
1 1 i∗ r
U (A)∗ U A U (A∗ A)∗ U (A∗ A)
∼
=
∗
U (A) U A k
∼
=
1 1 η
/
∗
U (A) U A
Z
1 1 copr
B
U (B)∗ U B
86
BRIAN DAY AND CRAIG PASTRO
The region labelled by (1) commutes on composition with 1 ⊗ n ⊗ 1 since
/
n
k
U A ⊗ U (A)∗
1⊗n⊗1
U A ⊗ U A ⊗ U (A)∗ ⊗ U (A)∗
n
1⊗1⊗c
U A ⊗ U A ⊗ U (A)∗ ⊗ U (A)∗
n⊗1⊗1
U A ⊗ U (A)∗
/
1⊗c⊗1
U A ⊗ U (A)∗ ⊗ U A ⊗ U (A)∗
commutes (choose a basis for U A). The region labelled by (2) now commutes by inspection
of
U A ⊗ U (A)∗ ⊗ U A ⊗ U (A)∗
O
1⊗σA ⊗1
1⊗c⊗1
QQQ
QQQ
QQQ
QQQ
QQQ
QQQ
QQ
/
U A ⊗ U (A∗ )∗ ⊗ U (A∗ ) ⊗ U (A)∗
O
1⊗c⊗1
∗
1⊗1⊗ρ⊗1
QQ
U A ⊗ U A ⊗ U (A)∗ ⊗ U (A)∗
O
∗
U A ⊗ U (A)∗ ⊗ U (A)∗∗ ⊗ U (A)∗
1⊗1⊗c
O
1⊗c⊗1
U A ⊗ U A ⊗ UO (A)∗ ⊗ U (A)∗
1⊗n⊗1
∗ ∗
(A ) ⊗ U (A)
QQQ U A ⊗ U (A ) ⊗ U
O
QQQ
QQQ
QQQ
1⊗u−1 ⊗u∗ ⊗1
QQQ
QQQ
(
VVVV
VVVV
VVVV
VVVV
1⊗ρ⊗1⊗1
VV
1⊗e∗ ⊗1
U A ⊗ U (A)∗
U A ⊗ U (A)∗∗ ⊗O U (A)∗ ⊗ U (A)∗
VVVV
VVVV
VVVV
VV*
/ U A ⊗ U (A)∗∗
1⊗1⊗c
⊗ U (A)∗ ⊗ U (A)∗ .
From the definition of the U -trace d(A) of A ∈ A , we have that
k
dim U I·(dim U A)−1
r0
/
k
r0
UI
U (d(A)−1 )
/ UI
ON ENDOMORPHISM ALGEBRAS OF SEPARABLE MONOIDAL FUNCTORS
87
commutes, so that the exterior of
U A ⊗ U (A)∗
:
tt
tt
t
tt
JJ
JJr
JJ
J$
1⊗u−1
U A ⊗ O U (A)∗
(n,r,r0 )
U (A ⊗
A∗ )
O
n
Un
kO
/
r0
UO I
U (d(A)−1 )
dim U I·(dim U A)−1
r0
k
/
UI
commutes.
Finally, the region labelled by (3) commutes on examination of the following diagram
∗
(U (A)∗ ⊗ 4 U A)∗ ⊗ U (A)
⊗ UA
U
UUUU
∗
−1
UUU(u⊗1)
UUUU ⊗(u ⊗1)
UUUU
UU*
iii
e∗ ⊗1⊗1iiiiii
i
iiii
iiii
k ∗ ⊗ U (A)∗ ⊗ U A
O
(U (A∗ ) ⊗ U A)∗ ⊗ U (A∗ ) ⊗ U A
OOO
(e,i,i0 )
1⊗u−1 ⊗1
OOO
'
1⊗c
k ∗ ⊗ U (AO ∗ ) ⊗
UA
O
OO
i∗0 ⊗rOO
i∗ ⊗r
O'
U (I)∗ ⊗ U (A∗ ⊗ A)
O 11
WWWWW U (e)∗ ⊗1
WWWWW
11
WWWWW
11
+
∗
∗
k ⊗ U A O ⊗ U (A)
1
OOO
(∗)
U (A∗ ⊗ A)∗ ⊗ U (A∗
11
−1
11
1⊗1⊗uO
OO'
1⊗U c
11
11
k ∗ ⊗ U A ⊗ U (A∗ )
11
OOO
11
i∗0 ⊗rOO
O'
111⊗U e
U (I)∗ ⊗ UO (A ⊗ A∗ ) 111
11
copr
11
1⊗dim U I·(dim U A)−1 ·n
11
1⊗U n
1
1⊗c
U (I)∗ ⊗ U I
(n,r,r0 )
O
1⊗U (d(A)−1 )
k∗ ⊗ k
i∗0 ⊗r0
/
U (I)∗ ⊗ U I
t:
tt
t
tt
tt
tt 1
t
tt
tt
U (I)∗ ⊗ U I
⊗ A)
copr
??
??
??
copr ??
?
/
Z
B
U (B)∗ ⊗ U B
whose commutativity depends on the hypothesis that (U, r, r0 ) is braided monoidal in
88
BRIAN DAY AND CRAIG PASTRO
order for
U A ⊗ U (A∗ )
r
/
c
U (A∗ ) ⊗ U A
r
(∗)
U (A ⊗ A∗ )
/
Uc
U (A∗ ⊗ A)
to commute.
4. The fusion operator
The unital von Neumann axiom on End∨ U implies that the fusion operator
f = (1 ⊗ µ)(δ ⊗ 1) : End∨ U ⊗ End∨ U → End∨ U ⊗ End∨ U
has a left inverse, namely g = (1 ⊗ µ)(1 ⊗ S ⊗ 1)(δ ⊗ 1). For this we consider the following
diagram.
∨
End∨ U ⊗ End
QQ U
δ⊗1
/
(End∨ U )⊗3
QQQ
QQQ
QQQ
Q
1⊗δ⊗1
δ⊗1⊗1
δ3 ⊗1 QQQQQ
QQ(
(End∨ U )⊗4
1⊗µ
/
End∨ U ⊗ End∨ U
δ⊗1
1⊗1⊗µ
/
(End∨ U )⊗3
1⊗η⊗1
1⊗S⊗1⊗1
1
(End∨ U )⊗3 o
1⊗µ⊗1
(End∨ U )⊗4
Q
1⊗µ
End∨ U ⊗ End∨ U o
1⊗µ
QQQ
QQQ
QQ1⊗1⊗µ
QQQ
QQQ
QQQ
Q(
1⊗S⊗1
(End∨ U )⊗3
In particular f = (1 ⊗ µ)(δ ⊗ 1) is a partial isomorphism, i.e., f gf = f and gf g = g.
5. Examples of separable monoidal functors in the present context
Unless otherwise indicated, categories, functors, and natural transformations shall be
k-linear, for k a field of characteristic 0.
For these examples we recall that a (small) k-linear promonoidal category (A , p, j)
(previously called “premonoidal” in [2]) consists of a k-linear category A and two k-linear
functors
p : A op ⊗ A op ⊗ A → Vect
j : A → Vect
ON ENDOMORPHISM ALGEBRAS OF SEPARABLE MONOIDAL FUNCTORS
89
equipped with associativity and unit constraints satisfying axioms (as described in [2])
analogous to those used to define a monoidal structure on A . The notion of a symmetric
promonoidal category (also introduced in [2]) was extended in [4] to that of a braided
promonoidal category.
The main point is that (braided) promonoidal structures on A correspond to cocontinuous (braided) monoidal structures on the functor category [A , Vect]. This latter
monoidal structure is often called the convolution product of A and Vect and is given
explicity by the coend formula
Z a,b
p(a, b, c) ⊗ f a ⊗ gb
(f ∗ g)(c) =
in Vect. The unit of this convolution product is given by j.
5.1.
Example. Let (A , p, j) be a small braided promonoidal category with
A (I, I) ∼
=k
where
j = A (I, −),
and suppose that each hom-space A (a, b) is finite dimensional. Let f : A → Vectf be a
very weak bialgebra in the convolution [A , Vect] so that we have maps
µ:f ∗f →f
and
η:j→f
δ :f →f ∗f
and
: f → j,
and
satisfying associativity and unital axioms, plus the very weak bialgebra axiom. Suppose
also that A ⊂ C where C is a separable braided monoidal category, with
p(a, b, c) ∼
= C (a ⊗ b, c) and j(a) ∼
= C (I, a)
naturally, and suppose the induced maps
Z c∈A
p(a, b, c) ⊗ C (c, C) → C (a ⊗ b, C)
are isomorphisms (e.g., A monoidal). We also suppose that each a ∈ A has a dual
a∗ ∈ A .
Define a functor U : C → Vect by
Z a∈A
UC =
f a ⊗ C (a, C);
then, by the Yoneda lemma, U (a∗ ) ∼
= U (a)∗ if f (a∗ ) ∼
= f (a)∗ for a ∈ A . Furthermore, by
the Yoneda lemma,
Z
a∈A
UI =
f a ⊗ C (a, I) ∼
= fI
90
BRIAN DAY AND CRAIG PASTRO
so that, by our assumption A (I, I) ∼
= k the maps η and induce respectively maps
r0 : k → U I
i0 : U I → k.
and
Maps r and i are described in the following diagram.
Z a,b
∼
=
/
U C ⊗O U D
f a ⊗ f b ⊗ C (a, C) ⊗ C (b, D)
O
Z
r
C separable
a,b
f a ⊗ f b ⊗ C (a ⊗ b, C ⊗ D)
i
Z
a,b
Z
fa ⊗ fb ⊗
c
∼
=
p(a, b, c) ⊗ C (c, C ⊗ D)
O
µ
U (C ⊗ D) o
Z
1
c
δ
f c ⊗ C (c, C ⊗ D)
These then produce a braided monoidal and comonoidal structure on U . Moreover, we
have i0 r0 = dim U I · 1 if and only if I ηI = dim f I · 1, and if f is a separable very weak
bialgebra, then U is separable since ri = 1 if µδ = 1.
Therefore, Theorem 3.1 may be applied when A and U satisfy the “U -irreducibility”
and “U -trace” criteria.
5.2. Example. Suppose that (A op , p, j) is a small braided promonoidal category with
I ∈ A such that j ∼
= A (−, I) and with each x ∈ A an “atom” in C (i.e., an object
x ∈ C for which C (x, −) preserves all colimits) where C is a cocomplete and cocontinuous
braided monoidal category containing A and each x ∈ A has a dual x∗ ∈ A . Suppose
that the inclusion A ⊂ C is dense over Vect (that is, the canonical evaluation morphism
Z a
C (a, C) · a → C
is an isomorphism for all C ∈ C ), and
Z z
∼
x⊗y =
p(x, y, z) · z
(naturally in x, y ∈ A )
so that
Z
z
C (a, x ⊗ y) = C (a,
p(x, y, z) · z)
Z z
∼
p(x, y, z) ⊗ C (a, z) since a ∈ A is an atom in C ,
=
∼
= p(x, y, a)
by the Yoneda lemma applied to z ∈ A .
ON ENDOMORPHISM ALGEBRAS OF SEPARABLE MONOIDAL FUNCTORS
91
Let W : A → Vect be a strong braided promonoidal functor on A . This means that
we have structure isomorphisms
Z z
∼
Wx ⊗ Wy =
C (z, x ⊗ y) ⊗ W z and
k∼
= WI
satisfying suitable associativity and unital coherence axioms. Define a functor U : C →
Vect by
Z a
UC =
C (a, C) ⊗ W a.
Then, if we suppose that W (x∗ ) ∼
= W (x)∗ for all x ∈ A , we have
Z a
∗
C (a, x∗ ) ⊗ W a
U (x ) =
∼
= W (x∗ )
∼
= W (x)∗
∗
Z a
∼
C (a, x) ⊗ W a
=
= U (x)∗ ,
so that U (x∗ ) ∼
= U (x)∗ , and
Z
a
i0 : U I =
C (a, I) ⊗ W a
∼
= WI
∼
= k,
so that i0 r0 = 1 and r0 i0 = 1. Also there are mutually inverse composite maps r and i
given by:
Z x,y
∼
r : UC ⊗ UD =
C (x, C) ⊗ C (y, D) ⊗ U x ⊗ U y
Z x,y
∼
C (x, C) ⊗ C (y, D) ⊗ W x ⊗ W y
=
Z x,y
Z z
∼
C (x, C) ⊗ C (y, D) ⊗
C (z, x ⊗ y) ⊗ W z
=
Z z
∼
C (z, C ⊗ D) ⊗ W z
=
∼
= U (C ⊗ D),
which uses the assumptions that C is cocontinuous monoidal and A ⊂ C is dense. Thus,
ri = 1 and ir = 1 so that U is a braided strong monoidal functor.
92
BRIAN DAY AND CRAIG PASTRO
5.3. Example. (See [6] Proposition 3.) Let C be a braided compact monoidal category
and let A ⊂ C be a full finite discrete Cauchy generator of C which contains I and is
closed under dualization in C . As in the Häring-Oldenburg case [6], we suppose that each
hom-space C (C, D) is finite dimensional with a chosen natural isomorphism C (C ∗ , D∗ ) ∼
=
∗
C (C, D) .
Then we have a separable monoidal functor
M
UC =
C (a, C ⊗ b),
a,b∈A
whose structure maps are given by the composites
M
UC ⊗ UD ∼
C (c, C ⊗ b) ⊗ C (a, D ⊗ d)
=
a,b,c,d
o
/
M
∼
=
M
∼
=
M
c=d
adjoint
C (c, C ⊗ b) ⊗ C (a, D ⊗ c)
a,b,c
C (a, D ⊗ (C ⊗ b))
a,b
C (a, (D ⊗ C) ⊗ b)
a,b
∼
=
M
C (a, (C ⊗ D) ⊗ b)
a,b
= U (C ⊗ D),
and r0 : k → U I the diagonal, with i0 its adjoint. Moreover
M
U (C ∗ ) =
C (a, C ∗ ⊗ b)
a,b
∼
=
M
∼
=
M
C (a∗ , C ∗ ⊗ b∗ )
a,b
C (a, C ⊗ b)∗
a,b
∼
= U C∗
for all C ∈ C .
5.4. Example. Let (A , p, j) be a finite braided promonoidal category over Setf with
I ∈ A such that j ∼
= A (I, −) and with a braided promonoidal functor
d : A op → Setf
for which each structure map
Z
u:
z
p(x, y, z) × dz → dx × dy
ON ENDOMORPHISM ALGEBRAS OF SEPARABLE MONOIDAL FUNCTORS
93
is an injection, and u0 : dI → 1 is a surjection. Then we have corresponding maps
Z z
/
k[p(x, y, z)] ⊗ k[dz] o o /
k[dx] ⊗ k[dy]
and
k[dI] o
o
//
k[1],
where k[s] denotes the free k-vector space on the (finite) set s, in Vectf . Define the
functor U : C → Vectf by
Z x
f x ⊗ k[dx]
Uf =
for f ∈ C = [k∗ A , Vectf ], with the convolution braided monoidal closed structure, where
k∗ A is the free k-linear category on A so that
Z y
Z x
gx ⊗ k[dy]
f x ⊗ k[dx] ⊗
r : Uf ⊗ Ug =
Z x,y
∼
f x ⊗ gy ⊗ (k[dx] ⊗ k[dy])
=
Z z
Z x,y
f x ⊗ gy ⊗
k[p(x, y, z)] ⊗ k[dz]
Z z Z x,y
∼
f x ⊗ gy ⊗ k[p(x, y, z)] ⊗ k[dz]
=
Z z
=
(f ⊗ g)(z) ⊗ k[dz]
Z z
=
U (f ⊗ g)
and
Z
i0 : U I =
x
k[A (I, x)] ⊗ k[dx]
∼
= k[dI]
k[1] ∼
= k.
Hence i0 r0 = dim U I · 1 = |dI| · 1. Thus, U becomes a braided separable monoidal functor.
5.5. Example. Let A be a finite (discrete) set and give the cartesian product A × A
the Setf -promonoidal structure corresponding to bimodule composition (i.e., to matrix
multiplication). If
d : A × A → Setf
94
BRIAN DAY AND CRAIG PASTRO
is a braided promonoidal functor, then its associated structure maps
X
X
p((x, x0 ), (y, y 0 ), (z, z 0 )) × d(z, z 0 ) =
A (z, x) × A (x0 , y) × A (y 0 , z 0 ) × d(z, z 0 )
z,z 0
z,z 0
∼
= A (x0 , y) × d(x, y 0 )
→ d(x, x0 ) × d(y, y 0 ),
and
X
j(z, z 0 ) × d(z, z 0 ) =
X
∼
=
X
z,z 0
A (z, z 0 ) × d(z, z 0 )
z,z 0
d(z, z)
z
→ 1,
are determined by components
d(x, y 0 ) d(x, y) × d(y, y 0 )
d(z, z) 1
which give A the structure of a discrete cocategory over Setf .
Define the functor U : C = [k∗ (A × A ), Vectf ] → Vectf by
M
Uf =
(f (x, y) ⊗ k[d(x, y)]).
x,y
Then we obtain monoidal and comonoidal structure maps
U (f ⊗ g) o
r
UI o
r0
/
i
i0
/
Uf ⊗ Ug
k∼
= k[1]
from the canonical maps
M
f (x, z) ⊗ g(z, y) ⊗ k[d(x, y)]
x,y,z
o adjoint /
M
z=u=v
M
f (x, u) ⊗ k[d(x, u)] ⊗
g(v, y) ⊗ k[d(v, y)]
x,u
and
M
v,y
k[d(z, z)] k ∼
= k[1].
z
These give U the structure of a separable braided monoidal functor on C .
ON ENDOMORPHISM ALGEBRAS OF SEPARABLE MONOIDAL FUNCTORS
95
6. Concluding remarks
If the original “fibre” functor U is faithful and exact then the Tannaka equivalence (duality)
Lex(C op , Vect) ' Comod(End∨ U )
is available, where Lex(C op , Vect) is the category of k-linear left exact functors from C op
to Vect. (See [3] for example.) Thus, since C is braided monoidal, so is Comod(End∨ U )
with the tensor product and unit induced by the convolution product on Lex(C op , Vect);
for convenience we recall [3] that, for C compact, this convolution product is given by the
restriction to Lex(C op , Vect) of the coend
Z
C,D
Z
C
F C ⊗ GD ⊗ C (−, C ⊗ D)
F ∗G=
∼
=
F C ⊗ G(C ∗ ⊗ −)
computed in the whole functor category [C op , Vect]. Moreover, when U is separable
monoidal, the category Co(End∨ U ) of cofree coactions of End∨ U (as constructed in [7]
for example) also has a monoidal structure (Co(End∨ U ), ⊗, k), this time obtained from
the algebra structure of End∨ U . The forgetful inclusion
Comod(End∨ U ) ⊂ Co(End∨ U )
preserves colimits while Comod(End∨ U ) has a small generator, namely {U C | C ∈ C },
and thus, from the special adjoint functor theorem, this inclusion has a right adjoint.
The value of the adjunction’s counit at the functor F ⊗ G in Co(End∨ U ) is then a split
monomorphism and, in particular, the monoidal forgetful functor
Comod(End∨ U ) → Vect,
which is the composite Comod(End∨ U ) ⊂ Co(End∨ U ) → Vect, is a separable braided
monoidal functor extension of the given functor U : C → Vect.
References
[1] Aurelio Carboni. Matrices, relations, and group representations, J. Algebra 136 no.
2 (1991): 497–529.
[2] Brian Day. On closed categories of functors, in Reports of the Midwest Category
Seminar IV, Lecture Notes in Mathematics 137 (1970): 1–38.
[3] Brian J. Day. Enriched Tannaka reconstruction, J. Pure Appl. Algebra 108 no. 1
(1996): 17–22.
96
BRIAN DAY AND CRAIG PASTRO
[4] B. Day, E. Panchadcharam, and R. Street. Lax braidings and the lax centre, in Hopf
Algebras and Generalizations, Contemporary Mathematics 441 (2007): 1–17.
[5] Brian Day and Ross Street. Quantum categories, star autonomy, and quantum
groupoids, in Galois Theory, Hopf Algebras, and Semiabelian Categories, Fields Institute Communications 43 (2004): 187–226.
[6] Reinhard Häring-Oldenburg. Reconstruction of weak quasi-Hopf algebras, J. Algebra
194 no. 1 (1997): 14–35.
[7] André Joyal and Ross Street. An Introduction to Tannaka Duality and Quantum
Groups, Lecture Notes in Mathematics 1488 (Springer-Verlag, Berlin, 1991): 411–
492.
[8] G. M. Kelly. Basic concepts of enriched category theory, London Mathematical Society Lecture Note Series 64. Cambridge University Press, Cambridge, 1982. Also at
http://www.tac.mta.ca/tac/reprints/articles/10/tr10abs.html
[9] Ross Street. Fusion operators and cocycloids in monoidal categories, Appl. Categ.
Struct. 6 (1998): 177–191.
[10] Kornél Szlachányi. Finite quantum groupoids and inclusions of finite type, Fields
Inst. Comm. 30 (2001): 393–407.
Department of Mathematics
Macquarie University
New South Wales 2109 Australia
Research Institute for Mathematical Sciences
Kyoto University, Kyoto 606-8502 Japan
Email: craig@kurims.kyoto-u.ac.jp
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ftp://ftp.tac.mta.ca/pub/tac/html/volumes/22/4/22-04.{dvi,ps,pdf}
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