2.1 EXAMPLE Write FDHILB for tre full subcategory of HILB whore objects ... and wInse morphians are tre moroidal fuoctors. ...... PRIX)F In fact, V X E Db g,.
RECONSTRUCTION THEORY
Garth Warner Department of Mathematics University of Washington
Suppose that G is a canpact group. objects are the continuous fini te
Denote by Rep G the category whose
d~nsional
unitary representations of G am
whose ItOrphisns are the intertv.rining operators -- then Rep G is a ItOnoidal *-category with certain properties Pl,P2' ... .
Conversely, if C is a ItOnoidal
*-category possessing properties Pl ,P 2 , ..• I can ore find a canpact group G, unique up to isarorphisn, such that Rep G "is"
£?
The central conclusion of
reconstruction theory is that the answer is affirmative.
CONTENTS § 1..
MONOIVAL CATEGORIES
§2.
MONOIVAL FUNCTORS
§3.
STRICTIFICATION
§4.
SYMMETRY
§5.
VUALITY
§6.
TWISTS
§7.
*-CATEGORIES
§8.
NATURAL TRANSFORMATIONS
§9.
CONJUGATES
§lO.
TANNAKIAN CATEGORIES
§ll.
FIBER FUNCTORS
§12.
THE INTRINSIC GROUP
§13.
CLASSICAL THEORY
1.
MONOIVAL CATEGORIES
§l.
Given categories
~,~,
their product is the category Ob(~ x ~)
=
~
x
~
defhed by
Ob ~ x Ob ~
MorUX,Y), (X' ,Y'» = Mor(X,X') x Mor(Y,Y')
with composition (f' ,gIl Now take ~
nmctor
e:~ x
= ~ --
£ -+
~
0
(f,g)
= (f'
f,g'
0
0
g).
then a nonoidal category is a category
~
equipped with a
(the multiplication) and an object e E Ob
£
(the unit), together
with natural isorrorphisms R, L, and A, where ~:X
e
e
-+
X
Ix:e e X
-+
X
and Ax,y,z:X e (Y e z)
-+
(X e Y) Q Z,
subject to the following assumptions.
X
e
A (Y & (Z Q W»
id III A
A
-+ (X & Y) & (Z & W) -+ «X & Y) & Z) & W
1
r
Alii id
X & {(Y & Z) & W) - - - - - - - - - - ' > {X & (Y & Z» A
cannutes.
& W
2.
A X Q (e Q Y) -+ (X Q e) Q Y
id 9 L
1
1R 9 id
XQY
XQY
corcm:utes.
[Note:
The "coherency" principle then asserts that "all" diagrams built up
fran instances of R, L, A (or their inverses), and id by repeated application of necessarily carn:nute.
In particular, the diagrams
A e Q (X Q Y) -+ (e Q X) Q Y
Ll
lL Q id
XQY
connute and L e
N.B.
= Re :e
A X Q (Y Q e) -+ (X Q Y) Q e
id
XQY ~
e
-+
~ Rl
lR
X~Y
X~Y
e.]
Technically, the categories
(~ x~)
x ~
are not the same so it doesn't quite make sense to say that the functors
-
c
(x,(Y,Z»
-+
X ~ (y
(f,(g,h»
-+
f
~
Z)
x (~ x~) -+ C ~
(g Q h)
Q
3.
«X,Y),Z)
-+
(X
~
Y)
~
Z
«f,g),h)
-+
(f
~
g)
~
h
(C x C) x C -+ C
are naturally isooorphic.
However, there is an obvious is::m::>rphism 1
-
C x (C x C) -+ (C x C) x C
-
am the assumption is that A:F
-+
G
-
is a natural iSClf1Orphisn, where
1
0
-
-
F C x (C x C) -+ C
-
(g
It x
-
g)
-
x
c
-+
c.
G -
Accordingly,
v (X,(Y,Z»
E
(g
Ob C x
g)
x
and
(g
v (f,(g,h}) E Mor C x
X
g),
~
Z
the square
Ax,y,Z x~
f
~
~
~
Z)
)0
(X
(Y'
~
Z')
)0
(X'
Ax',Y',z' comnutes.
Interchange Prin:::iple
~
Y)
1
1
(g ~ h)
x'
(Y
If f E Mor(X,XI)
g E Mor(Y,Y') ,
~
(f
Y')
~
~
Z'
g) ~h
4.
then (f fib id
)
(i~ fi1I
0
g)
=f
=
fib 9
(id
Y'
Since fib: g x
[Note:
(f
1.1
EXN1PLE
0
fi1I g)
0
(f
fib
i e
I
II ride
s
~
e
5
~
e
e~e
> e
5
~
e
s
t
e
e~e
ide \11
11\ ide
t
\ t
;::;
~
'> e
~
e
e~e
'> e.
e~e
Then -1 o (s R
0
e
t)
0
= R-e 1
R
e
o (t
0
s)
0
R
e
==> sot = t
Given f E Mor(X,Y) an:1 s E
~(~)
L
I
LEMMA
s.
define s·f to be the ca:nposition
-1
s
X --> e
1.5
0
~
~
f
L
X --> e
~
Y -> Y.
We have
id ·f e
=
f
s· (tof)
:::
(s o t).f (t o s). (g
(t·g)
0
(s· f)
(s· f)
~
(tog) == (s
o
t).(f
0
f)
~
g).
7.
A monoidal category
g is
said to be strict if R, L, and. A are identities.
So, if C is strict, then x ~ (y ~ Z) = (x ~ y) ~ Z
am X~e=X
e [Note: N.B.
While momidal, reither
x
X.
=
~
mr HIlB is strict monoida1.]
Take C strict am corsider ~(g)
f
1.6
~
EXAMPIE
~
g
then
= fog = 9
V
f = g
0
E ~(g),
f,g ~
f.
Let $ be the category whose objects are the nonregative integers
am whose morphisms are specified by the rule
% if n
~
m
Mer (n,m) = $n if n
= m,
romposi tion in Mer (n, n) being group multiplication in $. n
on objects by ~(n,m)
=n
+ m
and on mo:rphisms by
a L ~ (n -+ n, m -+ m)
=
P
n,m
(O,L),
Define
8.
is the canonical map, i. e. , -
p
n,m
1
2
n
n
+ 1
GX
1Gf ';:::
-y
>GY
[s:,S:1
is a
12.
;:;0'
~X'
F'X'
1
F'f'
1 G'f'
F'Y'
In particular:
G'XI
::>
> G'Y'.
The diagram H
F'X FF'X
GF'X
::>
FOxl
1Gox
FG'X
GG'X
::> ~
G'X camnites.
This said, too claim is Hat
E Nat(F
~ ~;:;O'
Q
F', G
Q
i. e., that too diagram (~ G ~')X
FF'X
::>
GG'X
IGG'f
FF'fl FF'Y
'>
GG'Y
(~ G ~')Y
corrmutes.
In
fact,
= GG'f
Q
GG'f
Q
=
-
F;:;O'
0
~X
G'X
G;:;O'
-x
0
....
FiX
GI ),
13.
G(G'f
=
~X)
0
~
0
F'X
=
F'f)
G{~y 0
0
_
F'X
= G~y
GF'f
0
0
H
F'X
= G;::;' -Y
0
H
o
FF'f
0
FF'f
F'Y
=-
o F~y
G'Y
=
1.10
LEMMA Suppose that
(~ ~ ~')Y
0
FF'f.l
S is :rronoidal
and let e,e' be units -- thm e
e' are i SJITOrphic. [T1:Ere is an i SJITOrphi s:n ¢: e
e' for which t1:E diagrams
-+
id 9 ¢ Xge
¢ 9 id
> X 9 e'
R,cl
e9X
,
lRX
X
> e' 9X
1'1(
rxl
X
X
X
comnute, viz. -
are iSJIl'Orphisns, natural in X,Y, subject to (MF1)
~ y)
F(X
tre fo11()"".N'ing- assumptions.
Tre diagram id
FX \21' (FY
A (FX
Fe is an i ::orrorphi su, and tl:e
~,
~,
~
=
~
> FX
FZ)
~I
~
F (Y
Z)
F(X
~,
(Y
~
Z»
1
1
FY)
~
FA
> F{X
FZ
=:
\21
~
Y)
~I
FZ
F ({X
id
~
Y)
~
Z)
~
carmutes. (MF2)
Tre diagrams
~ FX ~' e'
id
~
t;
e'
>FX
1
!~ > F{X
FX ~' Fe
~
e)
t;
~
id
LFx
~,
FX - - - - '> FX
1
Fe~'
!FIx FX - - - '> F{e ~ X)
-
~
cormrute. N.B.
2.1
A rroroida1 fur:ctor is said to be strict
EXAMPLE
1;;
am =: are identities.
Write FDHILB for tre full subcategory of HILB whore objects
are finite dimensional -- then the forgetful fur:ctor
2.
is strict rronoidal. [Take for
tffi identi ty i~ & y and let t; [Note: E.g.:
= ide']
A forgetful functor need not be rroroidal, let alone strict rroroidal.
Give AB its rronoidal structure per the tensor proouct, give SET its rron-
oidal structure :per the cartesian proouct, and consider U:AB
+
SET -- then tm
canonical maps '(]A
x DB + U(A
&z
B)
are not iSOIDrphisrns.]
Let (F , ~,:::)
(G,e,G) be rronoidal functors -- tmn a rronoidal natural transformation (F,~,~)
is a natural transfonnation a:F a Fe
~
+
+
(G,e,G)
G such that tm diagrams
e FX &1 FY
>Ge
i
ie
e'
el
,
~ Q.I
-
1
CYyl
ax &
GX 9. 1 GY
> G{X
G
carmute.
> F{X & Y)
Q.
Y)
y
3.
ViTrite [g,g']~ for the netacategor:y whose objects are the m::noidal ftmcmrs g
-+
g' and whose rrorphisms are the m::noidal natural transfonnations. N.B.
A m::noidal natural transfonnation is a m::noidal natural isorrorphism
if a is a natural ia::m::>rphism.
2.2
Sane authorities assmne outright that Fe
REMARK
= e',
t1:e rationale
being that this can always be achieved by replacing F E Ob [g,g']~ by an ia::m::>rphic F' E Ob [g,g']~ such that F'e = e' (on objects X
2. 3
I..EMt-1A
e, F'X = FX).
";!:
Let (F, 3,1;)
(F I ,3' , 1;' ) (F':g I
-+
be rroroidal fun::::tors -- t.1:en treir comp::>si tion F' 1;' [Consider the arrc:JINs e I I
--~>
F'FX ~I I F'FY
wri te
~4:NCAT
C' , )
0
F is a rronoidal functor.
F'I; FIe' -----> FIFe and
F'-
~I
- FX,FY> F' (FX ~I FY) _ _-_X-'-,_y::> F'F(X ~ y).]
for tre metacategory wInse objects are tre monoidal categories
and wInse morphians are tre moroidal fuoctors.
2.4
RAPPEL
Let g, 12 be categories -- then a fun::::tor F:g
an equivalence if there exists a functor G:12 FOG ::::
i~,
-+
g such that G
the symbol:::: starrling for natural iEOlrorphian.
0
-+
12 is ::aid to be
F
~
id
c
and
4.
2.5
A furr:tor F:g
LEMMA
-+ ~
is an equivalence iff it is full, faithful,
and res a representative ima.ge (Le., for any Y E Db
~,
tlere eKists an X E Ob C
such tret FX is iEa1X)rpm.c to y) .
Categories
N.B. aleoce F:C
-+
g,
~
are said to be equivalent provided ther-e is an e::ruiv-
The abject is:ttOrphisn types of equivalent cate:.:rories are in a
D.
one-to-one corresfOrrlerr:e.
F:C
-+
D
G:D
-+
C
from cOP x
12
C
2.6
RAPPEL
Given categories
, functors
are said to be an
D
Mor
0
(FOP x i~)
adjoint pair if the functors Mor
0
(id OP x
to SET are :naturally
~)
C X E
Db C
Y E
Db D
isonorphic, i.e., if it is possible to assign to each ordered pair
a bijective nap E!x,y:M::>r{FX,y)
-+
this is so, F is a left
---"""'----
M::>r{X,GY) which is functorial in X and y.
for G and G is a right adjoint for F.
left (right) adjoints for G (F) are raturally isarorphic.
When
Any UNo
In order that (F ,G)
be an adjoint pair, it is recessary am sufficient that there exist natural trans~ E
c,
Nat(id
G
0
F)
(VF)
0
(FV)
=
(Gv)
0
(~G)
= idG•
i~
subject to
formations
v E Nat(F
0
G,
i~)
The data (F ,G,~, v) is referred to as an adjoint situation, the natural trans-'
5.
fonnations
being the arrOW'S of adjunction.
v:F
0
G
An
adjoint eg;uiv-
-+ i~
alence of categories is an adjJint si tllation (F ,G,l1, v) in which roth 11 and v are ffi tural
isorrorphisrns.
2.7
W1MA
A ftlllctor F:C
-+
D is an equivalence iff F is prrt of an adjoint
eg;uivaler:ce.
Let
g, g'
be rronoidal categories
g,
then
C' are rroroidally equivalent
if there are moroidal functors F:C
C'
-+
F' :C'
-+
C
and rroroidal natural isorrorphisms
F
0
F' - id
C'
2.8
LEMMA
-r~'
Suppose that F:g
is a rroroidal fuoctor.
AsSUIllE!:
equivalence -- then F is a rronoidal equivaler:ce.
2. 9
REMARK Embe:i F in an adjoint si tlla tion CF, F' ,11,11 '), where
l1':F
0
F'
-+
id
C'
F is an
6.
are the arrows of adjunction (cf. 2.7) -- then one can equip F' with the structure of a nnnoidal functor in such a way that the natural isonorphisms 'fl, 'fl' are Thus first specify ~':e
nnnoidal natural isonorphisms.
F,~-l F'Fe ----'> F'e'.
be the composition e
~,
x' ,yl
:F'X'
~
-+
F'e' by taking it to
As for
Fly'
-+
FleX'
~'
Y'},
build it in three stages:
1.
FIX'
2.
F'F(F'X I
~
Fly'
~
'fl -+
F'F(F'X'
~
F'Y'};
F'Y'} --------> FI(FFIX'
~'
FF'Y'};
'fl' Xl 3.
FF'X'
---~
X'
'fl' y' FF'Y'
> y'
=>
'fll
~
X'
'fl' :FF'X'
~'
FF'Y'
-+
X'
~
, Y'
Y'
=>
F' ('fl'
X'
~
'fl' ) :F' (FF'X'
~'
FF'Y')
-+
F' (X'
~'
Y').
Y'
If C is m::moidal, then ~OP is m::moidal when equipped with the sane ~ and e, taking
1.
§3.
STRICTIFICATI0N
A strictification of a rroroidal category which is rroroidally equivalent to
3.1
EXAMPLE
~
~.
MATJc is a strictification of
-
[Tte equivalence MATk -+
is a strict rroroidal category
FOVE'.C1c
F'DVE fFT
1
1~
Frs
> FfT,
Ff
5.
where f E Mer (rS, rT) corresponds to u. [Note:
COJ:tposi tion of Il.Dl1oidal fl.IDctors is preEerved by this construction.]
There are five ingredients figuring in the definition of a Il.Dl1oidal categor.y: S, e, R, L, A.
Keeping track of R, L, A in calculations can be annoying and one
way out is to pass from £: to £:str' a IIDre complicated entity than
g.
But this too has its downside since ~tr is So, in what follows,
'We
shall stick with £: and
detennine to what extent R, L, A can be eliminated from consideration (Le., are identi ties) • Suppose that
(S, E, R, L, A) (S',
are IIDnoidal structures on
£ --
Therefore t;::e'
-+
R', L', A')
then these structures are deem::d isorrorphic if :I a
Il.Dl1oidal equivalence of the fonn N.B.
e',
(idc'~'~)
between them.
e is an isorrorphism and the ~X,y:X
9' Y -+ X S Y
are isorrorphisms, subject to the coherence conditions of §2.
3.4
REMARK
The philosophy is that replacing a given Il.Dl1oidal structure on £
by another isorrorphic to it is of no consequence for the l.IDderlying ma.thematics.
3.5
LEMMA let (S, e, R, L, A) be a Il.Dl1oidal structure on
a ma.p S' :Ob
~~-
...- - - - -
g
x Ob
g -+
Ob
g,
g.
Suppose given
an object e' E Ob £:, an iSOllOrphism t;:':e
-+
e', and
6.
isarorphisrns
(~',
Then there is a unique rronoidal structure
(id ' C
~',
~I,
E'):(£,
e', R', L', AI) on C such that -+ (£,~,
e ' , R', L', A')
e, R, L, A)
is an iEOJ.rorphisn. Extend~'
PR(X)F
£
to a functor ~':C x
-+
£ by
the prescription
;::;1
-X,Y ~(X,Y)
~(f,g)
>
~'
1~I
1
~(X'
(X,Y)
>
,Y')
~'
(f,g)
(X' , y' ) ,
;:;1
x' ,Y' s:> ~ ::;::~'
(via;::;' E
Nat(~,~'».
This done, define R', L', AI by the diagrams
R'
L'
X ~' e' E' X
> X
r ~
e
rR e' < id ~ ~'
l
;::;.
X~e
e
l
~'
> X
X
I
I
~X
(X
Z)
A
~
Y)
~
L
Z.
7.
3.6
THEOREM let
(~,
e, R, L, A) be a rronoidal structure on C.
e 1 is an object is::xrorphic to e, say ';:e' (~',
rronoidal s tructure
e
t,
then there is an isorrorphic
e -
-+
g in which
R', L t , A') on
Suppose that
R', L t are identities.
PRl:X>F Bearing in mind 3.5, put X~'
Y
=X~
Y if X
~
e'
~
Y
and
X~' Y
=e
Y if X
-
1
=
XifY=e'. Define
by stipulating that
EX,y is to be the -~,
~
identity if X
= ~
0
(i F{Y 2 X)
ccmnutes. N.B.
The nonoidal natural transfo:r:ma.tions between syrmetric n:onoidal functors
are, by definition, "syrmetric n:onoidal lt (Le., no further conditions are imposed
5.
that reflect the presence of a synJlEtry) • [Note:
Therefore the su1::x::atego:ry
[g,g'] ~ , T
of
[g,g'] ~
whose objects are the
SynJlEtric nonoidal natural transfo:rma.tions is, by definition, a full su1::x::atego:ry.]
4.5
Recall that
EXAMPLE
~n
has the following presentation:
It is generated
by 0 ,." ,0 _ subject to the relations 1 n l
Suppose now that
1
0~1 =1 1,0,0'+10. = 0'+10,0'+1,0.0. = 0,0. 1 1 111 1 J J 1 1).
g
Define auto-
is synJlEtric strict lIDnoidal and fix X E Ob
~-l
lIDrphisms II , ••• , II
g.
Qn
of X-- by
Then there exists a unique horrorrorphism
of groups such that (i
Combining the
4. 6
rrxn then
leads to a synJlEtric nonoidal functor F:~
LEMMA let F: g
then the sy.rmet:ry
T
= I, ... ,n-l).
on
-+
g'
g can
be a nonoidal equivalence. be transferred to a sy.rmet:ry
as to render F sy.rmetric nonoidal. [Define TFX,FY by
-+
Assl.lIIe: T'
on
c'
C such that
g is SynJlEtric in such a way
6.
FT FX ~' FY - > F (X ~ Y) - > F (Y ~ X) - > FY ~' FX M
and recall that F has a representative image (cf. 2.5).]
4 •7
and y: C
-
EXAMPLE
-+ C
4.8
-s
(cf. 2.9).
~
is syt.l1:'OOtric rronoidal, then
~str
is syr:t:1lIetric rronoidal
tr is a symnetric IIDnoidal equivalence.
LEMMA
equivalence.
If
-
ret C, C' be symnetric nonoidal and let (F,F' ,fl,fl') be an adjoint
Assurre:
...
F is sy:rmetric IIDnoidal -
then F' is syrmetric rronoidal
1.
§5.
let
g be a oonoidal category -
VUALITY
then each X E Ob
- e x:c
-+
C
x e -:c
-+
c.
-
Definition:
g defines flIDctors
-
C is
left closed righ t cloSErl if
vXE
Ob
g,
- e X [Note:
g is
X admits a right adjoint, denoted lham(X,-)
e -. admits
a right adjoint, denoted rhom (X,-).
closed if it is berth. left closed and right closed.]
So:
g left closed ==> M::>r (Y ex, Z) :::: M::>r (Y , Than (X, Z»
g right
closed => M::>r(X 9 Y,Z) :: M::>r(Y ,rhom(X,Z»
for all Y,Z E db C. N. B •
The f'lIDctor lhom(X,-) rham(X,-) -left internal hom functor attached to X.
is called the right
2.
5.1
REMARK
If g is syIl'll'!etric rronoidal, then left and right internal hans
are naturally isomorphic and if
5.2
EXAMPLE
g is
left or right closed, then C is closed.
!S,
Given a ccmnutative ring
let MOl\. be the category whose
objects are the left !s-m::xlules and whose norphisms are the !s-linear maps ~
is symretric nonoidal.
5.3 -
~
I..ll~
then
M::>reover, 14:)1\. is closed and
lhomeX,Z} ::::
~
eX,Z)
rhom(X,Z) ::::
~
(X,Z).
Suppose that
g
is left closed -
then V X E Cb
g,
the flIDctor
X preserves colimits (being a left adjoint) and the flIDctor lhomeX,-)
preserves limits (being a right adjoint) .
5.4
LEMMA
Suppose that
g
is left closed -- then V Z E Cb
g,
the cofl.IDctor
Thorn (-, Z) converts colimi ts to limits. PRCX)F
let! be a small category, I'J.:!
exists -- then V Y E Cb
-+
g
a diagram for which
g,
:::: Mor(Y :::
~ col~
M::>recol~
-
(Y
l'J.i,Z} ~
l'J.i ) ,Z)
col~
l'J.i
3.
=>
Let
g
be a m:m.oidal category.
Given X E Ob
g,
an object
to be a left dual of X if :3 IlDrphism.s
and corrmutative diagrams L-l X
nx 6b id
--'>
------''>
(X 6b v X) 6b X
X 6 b e < - - - - X 6b ( vX 6b X) • id 6b E:
X< R
x
-1
Vx
e 6b X
R
'>
v id 6b X6be
nx
II
> Vx
e ex
6b vX)
lA
Vx r (y 6a X, Z) ;:::.Mer (y I Z 6a
v
X)
v .Mer ( X 6a Y I Z) ::::.Mer (y I X 6a Z) •
PR(X)F
It will be enough to sh:::M' that -
the proof that Vx 6a -
6a
Vx is a right adjoint for
is a left adjoint for X 9 -
being similar.
F=-6aX
(cf. 2.6) G=-6a
am to simplify the writirg, take
v
X
g strict.
v E Nat(F
0
Define
Glide)
So let
6a X,
5.
by
Pw E M:>r(W,W ~
Vw E M:>r(W ~
X ~
v
X)
v
X ~ X,W)
Consider (vF)
0
(F~).
Thus «vF)
(~»W
0
=
(VF)W
0
(~)W'
And ~
E Nat(F,FGF)
or still,
i W
~ X ~
vF E Nat(FGF,F) (V)
or still,
(vF)W:W ~ X ~
v
X ~ X - - - - - - - - ' : > W ~ X.
Vx ~
X.
6.
Therefore
is the cauposition (i~ 9 i~ 9 EX)
=
(i~
0
i~) 9
0
(i~ 9 nX 9 i~)
«i~ 9 EX)
0
(nx 9
i~))
=i~9i~
= i~9X =i~
= (i~)W·
I.e. :
= i~.
(vF)
0
(Fp)
(Gv)
0
(pG) = id G
The verification that
is analogous.
5.6
LEMMA A left dual of X, if it exists, is unique up to isorrorphisrn.
PR.CXJF
suppose that
are two left duals of X -- then the ftmctors
7.
_9v~
are naturally isarorphic (roth being right adjoints for -
NOw specialize and take W = e to get
=>
Vx
1
[Note:
Explicated,
L
-->
V
X o] 2
~ ~
vX'.... -""2
9 X), so 'if W E Ob
f,
8.
be an isarcorphism and put
[Consider first the case when
But
=>
£
is strict, thus, e. g. ,
9.
=>
In. general, the claim is that i X
~
(vX ~ X)
> X
~
v ( X ~ X).
(tj>-l ~ id)
is (X ~ vX ) ~ X
A
-1
However, due to the naturality of the associativity constraint, there is a
10.
conmutative diagram -1
A (X a v X ) ax
(id a
X
a (vX a X)
1
1 V X ')
id II
(~ II id)
> X a (vX' a X) •
ax -1
A
And (id a ( a id»-l
= id
a (-1 aid).]
A rronoidal catego:ry ~ is said to be left aut.onorrous if each object in ~
admits a left dual. N.B.
Suppose that ~ is left autoncm:::>us.
Given f E M:>r(X,y), define
-1
R
id a
fix
A
(id a f) a id
-------::>
L
--.....;>
v X.
e
9
V
X
11.
'Iben the assignnent
defines a cofunctor C [Note:
5.8
-+
C.
The specific form of vf depends on the choices of
REMARK
If C is left autonarrous and if X, Y E Ob
g,
Vx and
vY .]
then v eX 9 Y) is
[We have M::>r(V (X 9 Y) 9 W,Z) :::: M:>r(W, (X 9 Y) 9 Z) :::: M:>r(W,X 9 (Y 9 Z»
v
:::: Mor( X 9 W, Y 9 Z)
v
v
:::: M:>r e Y 9 ( X 9 W) , Z)
v
v
:::: Mor« Y 9
X) 9 W,Z)
=>
5.9
~
PRCX>F
SUpp:>se that
In fact, V X E Ob
g is left autonarrous -
then C is left closed.
g, lhan(X,-)
=
v
9 X.
(he can also introduce the notion of a right dual Xv of X, where this t.ine
12.
subject to the obvious aJIllllUtativity conditions.
Here the functor -- ~ XV is a
left adjoint for the functor -~ X and the functor XV for the functor X [Note:
e-
is a right adjoint
e -.
If X admits a left dual Vx and a right dual xv, then in general Vx
and XV are not isarrorphic.
the other hand, it is true that
On
E.g.:
=>
The definition of ttright autonOODus" is clear and
'WI9
shall tenn C autonooous
if it is roth left and right autOllOIIOUS.
5.10 PRIX)F
LEMMA
Suppose that
In fact, V X E Db
g is right autonOODUS -- then g is right closed.
g, rham(X,-)
5.11
variables.
REMARK
If
g is
= XV e -- .
autonarrous, then --
e-
preserves colirnits in roth
13.
Suppose that F:£ of X -
£' is a rronoida1 ftmctor.
-+-
v then FX is a right dual of FX.
Assurre:
Xv is a right dual
Consider the arrows
Proof:
-1
---\>
e' [Note:
-\>
t; Fe - - - > e'
Fe
Assurre that
~, ~'
are right autonc:m::ms ~:FX
v
-+-
(FX)
v I
n.anely the canposition v FX
L ---\>
n 0 id _ _ _\>
e' 0' FX
v
«FX) v 0' FX) 0' FXv
A-I _ _ _> (FX) v 0'
id 0
(FX 0' FXv)
:=::
_ _----'> (FX) v 0' F{X 0 Xv)
id 0 Fe: _ _ _\>
(FX) v 0' Fe
id 0 t;-1 _ _ _\>
R
(FX) v 0 1 e' V
- - - > (FX) ,
and the diagram
then there is an iSCJIDrphisn
14.
FX 9' FX
id 9 ~
v
FX 9' (FX) v
:>
1
"1
EFX
F(X 9 Xv)
>
e'
FEX ccmnutes.] N.B.
Che can, of course, work equally well with left duals.
5.12
LEMMA
Let (F ,~,::n
(G,8,G)
be rronoidal functors and let o.:F -+ G be a nonoidal natural transfonnation. '!he source C of F and G is autonarrous PRCX>F
'!he claim is that 'If X E Ob
then
Fran the above, FX
v v FX (GX) is a left dual of FX (GX ).
v
GX
v (GX ) is a right dual of FX (GX) or still,
'Ihis said, fo:rm
and consider v
(0. )
X
is a nonoidal natural isarorphism.
g,
~:FX -+
is an isarorphism.
0.
AsS\.lrlE:
:GX -+ FX.
15.
[Note:
Accordingly,
g is autoncm::>us, then
the rretacategory
[g,gr]9 is
a groupoid.]
Suppose that C is syrt"Il'etric rocmoidal and left autoncm::>us -
-
autoncm:ms, hence C is autonooous.
Proof:
Given X E Ob
g,
then C is right
take XV
-
=
Vx and
define rrorphisms
V
e-+X
9X
by £
0
T
X
X,VX
T
0
x,vx
5.13
EXAMPLE
FDVECT is autonorrous. k
so it suffices to set up a left duality. V
nx .
In fact, FDVECT
k
'rhus given X, let Vx be its dual and define
~: X 9 X -+
!!
by ~(A'X)
en
= A(x) •
the other hand, there is a canonical isoonrphism
and we let
is syrmetric rronoidal,
16.
[Note:
5.14
An object X in ~ admits a left dual iff it is finite d.irrensional. J
EXAMPIE
The full su.1:::lcategory of ~ whose objects are finitely
generated projective is autonarous (cf. 5.2).
Assurre still that
5.15
LEMMA
£
is syrm:etric rronoidal and left autoncm:>us.
There is a rronoidal natural isorrorphism
ide -+ VV (_). [To see this, consider the composition
-1
R
X
-----e>
id
Q
X Q e
n
-----e>
(X Q (vX Q vvX )
A
_____ > (X Q v X ) Q vVx T Q id _ _ _ _ _;>
E:
(vX Q X) Q VVX
Q id
VV
-----e> e Q
L - - - - - e > VVX .]
N.B.
Let
X
17.
be the arrow constructed al:x::>ve -
then
(cf. 5.12).
But
v
here X
[Note:
= vX,
so
To make sense of this, recall that X is a left dual of XV
vv
X
.
lS
a left dual of
And
=>
v
(6
XV
):
vv
X -+ X.]
vv
v (X) •
1.
§6.
let
TWISTS
g be syrmetric rronoidal and left autonan:ous -
then a twist
n is a
nonoidal natural isan:orphism of the identity functor ide such that v x E Ob
[Note:
Tacitly, ide is taken to be strict
(~
= id,
E!
= id),
g,
tlms from the
definitions
'Ib consolidate the tenninology, a syrmetric rronoidal
g which is left autono-
nous and has a twist n will be referred to as a ribbon category. N.B.
'!he choice ~ == ~ is pennissible, in which case
It was pointed out near the end of autonan:ous.
§5
'Ibis fact is true in general.
by
-+ X
v
be even.
that an even ribbon category is right Proof:
and define norphisms
e
g is said to
9 X
Given X E Ob
g,
v
take X
= vX
2.
6.1
I...EM.I1A
PRCOF
In the presence of a twist Q,
Consider the canposition
A
_ _ _ _ _--,> (X
~
T
~ v X) ~ v (v X)
id
x,vX
(vX ~ X) ~ v (vX)
_ _ _ _ _-'?
_____> e L
~
v (vX)
v v
- -__> ( X).
E.g.: v
6.2
LEMMA
rrorphi sm f :X
let
-+ Y
g be
e::::
v
e
~
e::::
v
e
~
v v
(e)::::
v v
(e
~
e)::::
v v
(e):::: e.
In the presence of a twist Q, the left and right dual of every
agree:
vf -_ fV.
a ribbon category.
Given f E Mor(X,X)
I
define the trace of f by
3.
[Note:
LEMMA We have
6.3 1.
2.
trX(g
0
f) = try(f
0
g) (f:X -+ Y, g:Y -+ X):
3.
Put
the dinension of X.
So, on the basis of 6.3,
dim X
= dim Vx
and
dim (X N .B.
~ Y) =
Take!J = id -- then the categorical dimension of X is the arrow T
nX
e
6.4
(dim X) (dim Y) •
EXA1\1PLE
consider
then the trace of f:X
-+
> X ~
F'IJVF.Sc
x,Vx
Vx - - - - > Vx
EX ~ X --.....,> e.
(viewed as an even ribbon catego:ry (cf. 5.13»
X is the cartl[JOsition
-
4.
f k
--;:>
X ~ Vx
~
id
T
Vx
-----;:>
X,VX
X ~ Vx
-------:;:>
Vx ~ X
'Iherefore the abstract definition of trx(f) is the usual one.
--;:>
k.
In particular:
dim X = (~X)~.
E.g.: dim ~n = n~,
the distinction between n E N and
n~
being potentially essential if k has non-
zero characteristic.
6.5
REMARK While evident, it is important to keep in mind that the defin-
itions of trace and d..imension depend on all the lmderlying assumptions, viz. that our m::moidal g is syrmetric, left autoncm:ms, and has a twist
n.
Suppose that g, g' are rib1x>n categories wi th respective twists then a syrmetric m::moidal flmctor F:g
+
g' is twist preserving if
V
n, n' -X E Ob g,
F~= ~.
6.6
LEl-1MA
If F:C
+
C' is twist preserving, then V f E Mor(X,X), the diagram 1;
e' trFX(Ff)
;:>
1
1
e'
;:>
1;
carmutes.
Fe
Fe
Ftrx(f)
5.
Matters are invariably simpler if
£:
is a strict ribbon category, which will
be the underlying supposition in 6. 7 - 6.9 belOW'.
6. 7
LEM1A
'!he arrows
are mutually inverse isarorphisrns. PRCOF
Take X
=e
in the relation
to see that £
NOW' fix an isarrorphism X ~
e
id
T)X
e
:> X 0
id
Vx Vx
~
:> X ~
Vx
X
Vx
n
Vx :>
)
e
o T)
e
T)
o
=s
(s
o
(s
o
e
0
e
0
T)
e
7.
are equal, thus the compositions
id~nx
Vx ~ e
Vx
=
Vx
= Vx
are equal.
~
id~~~id
> Vx ~ X ~ Vx
;;. Vx ~ X ~ Vx
id ~ nx id ~ id ~ Qv e _ _ _ _> Vx ~ X ~ Vx _______ X_> Vx ~ X ~ Vx
Postcompose with EX ~ idvX -
then the first line gives v~, while
the second line is
or still,
or still, (id e
~
Qv )
X
(EX ~ id
0
Vx
)
0
idv
X
~
nx
or still, Q
Vx
6.10
REMARK
let
transferred to £str' noted already in 4.6.
where
E
~
oidV
X
=n ~bV'
X
be a ribbon category -- then this structure can be
'Ihat the synItl2!try
T
passes to a synmetry
T
str of £str was
As for the left duality, a generic element of £str is a
and n are defined in the obvious way.
It is also clear that the twist
8.
on £ can be brought over to a twist on £str"
Accordingly, y:C -
-+ C
-s
tr is a
syrmetric rronoidal equivalence which is twist preserving, i.e., y:C
-
a riboon equivalence"
-+ C
-s
tr is
1.
*-CATEGORIES
§7.
£
let :Js be a ccmnutative ring -- then a category r-t">r (X, Y) is a
~-m:.:xlule
is
~-enriched
and if the composition of norphisns is
functor F bebJeen :Js-enriched categories is
~-linear
if V X, Y E Ob
~-bilinear.
£,
A
if the induced maps
r-t">r (X, Y) + r-t">r (FX,FY) arehoIrom:::>rphisns of [Note:
functor
~:£ x
If
£
is :Js-enriched and nonoidal, then
£
+
£
is assuned to be
EXAMPLE
Suppose that
£
£
x
£
is :Js-enriched and the
~-bilinear.]
An object X in a ~-enriched category
N.B.
7.1
~-m:x1ules.
£
is irreducible if !-br(X,X) =
is Z-enriched and m::::noidal. :Js
~~.
Put
= ~(£) • £
'Ihen :Js is a unital ccmnutative ring (cf. 1.4) and
is
~-enriched
as a m::::noidal
category (cf. 1.5). [Note:
Suppose in addition that
£
is a ribl:::x::n category -
trx:M:Jr(X,x) + k is
~-linear
and V X,Y E Ob
~,
the map
M:Jr(x,Y)
1_ is
~-bilinear.]
~
M:Jr(Y,X) + k
f ~ g + trX(g
0
f)
then
V
X E Ob
£,
2.
A *-category is a pair (g,*), where g is a category enriched over the field of corrplex numbers and
is an involutive, identity on objects, positive cofunctor.
Spelled out:
V X,Y E Ob C, t-br(X,Y) is a canplex vector space, composition
t-br(X,Y) x t-br(Y,Z)
-+
t-br(X,Z)
is corrplex bilinear, *:t-br(X,Y)
-+
t-br(Y,X)
(zf + wg) *
= zf*
subject to
+ wg*
and
=f
f**
(g
f)* = f*
Q
0
g*.
Finally, the requirerrent that * be positive means:
[Note:
V X E Ob
g,
=0
f*
0
f
isitive, then A is a semisirnple algebra, hence "is" a multirra.trix algebra.]
let f: X f*
0
f
= i~
-+
Y be a rrorphism in a *-category
£ --
then f is an isometry if
and f is unitary if both f and f* are isametries.
let F be a C-linear functor between *-categories -- then F is *-preserving
=
if V f, F(f*) N.B.
(Ff)*.
Suppose that F is a *-preserving m:::m.oidal functor between nonoidal ~: e
then F is unitary if the isorrorphisms
*-categories -
'
-+
Fe and
are unitary. let p:X
-+
X be a rrorphism in a *-category
£ -- then p is a projection if
p = p* and pop = p. [Note: let
£
If g:Y
-+
X is an isometry, then g
be a *-category and let X,Y E Db
g*:X
0
£ --
-+
X is a projection.]
then X is a subobject of Y if
3 an isametry f E M:>r (X, Y) • ~finition:
£ has subobjects if for any Y E
q E M:>r (Y, Y), 3 X E Db Definition:
£
Ob
£ and any projection
and an isometry f E M:>r (X, Y) such that f
£ has direct sums if for all X, Y E
f E M:>r(X,Z), g E M:>r(Y,Z) such_ that f
0
f* + g
0
Ob
£,
g* = id •
z
0
f*
= q.
:3 Z E Ob C and isometries
5.
E.g.:
7.4
FDHIIB has subobjects and direct sums.
RAPPEL A category
£
is essentially sma.ll if
£
is equivalent to a SIl:\3l.l
category.
£
SUppJ5e that
is a *-category which is essentially small -- then
e
is
semisimple if the following conditions are net: SSl:
V X,Y E Ob
£,
dim MJr (X, Y)
FX E Ob g' irreducible and (b) X, Y E Db g irreducible and nonison:orphic => FX,FY E Ob g' irreducible and nonison:orphic.
1.
NATURAL
§8.
let
8.1
g,
TRANSFO~~ATIONS
C' be *-categories and let F:C -+ C' be a *-preserving functor.
LEMMA Nat(F ,F) is a unital *-algebra under the follO'lll7ing operations:
(a
0
B)
= a....
Xx
!3 X
0
[To check the *-condition, observe that \;/ f E M::>r(X(Y),
Ff
0
(a*)x
= Ff
(~)*
0
= (Ff*) *
8.2
EXAMPLE
Take
S' = FDHILB,
=
(~
=
(Ff*
0
(~)*
Ff*) *
0
ely) *
0
= (ely) *
0
=
0
(a*)y
put Na~
(Ff*) * Ff.}
= Nat(F ,F),
and let RePfd Natp
be the category whose objects are the finite dirtensional *-representations of
2.
Na ~ and whose norphisms are the intertwining operators.
Define a *-preserving
functor
as follows: (X E Ob
cpf
= Ff
(f
£)
M:>r(X,Y}).
Here
thus the diagram
FX
----------~~
FX
FY
- - - - -.....>
FY
conmu tes, so Ff is an intertwining operator. [Note:
If U:RePfd Nat F
is the forgetful functor, i.e., U(TI,H)
8. 3
THEOREM
let
functor.
Assl.lIle:
g
£, g
I
= H,
-+
then U
0
cP
= F.]
be *-categories and let F: g
-+
g I be a *-preserving
is semisirople -- then there is an isarorphism IfF:Nat(F,F)
-+
1T
iEIc of 'lIDi tal *-algebras.
FDHILB
M:>r(FX. ,FX.} 1
1
3.
PRCX)F
The definition of 'l'F is the obvious one:
'l'F (a)
= 1T . EI ~ e
~ .• ~
'l'F is injective: ~. = 0 ViE Ie => ~ = 0 V X E db
-
~
g.
To see this, chaos:! the Sik E Ivbr (Xi'X) as in the proof of 7.8 -- then
~=~oFi~
Bu t the diagram
~.
~
FX.
>
~
F (Sik)
FX.
~
1
F(Sik)
1 FX
>FX
~
conmu tes, hence
=
o.
'l'F is surjective: V {a. E Mor(FX.,FX.):i E Ie}' 3 a E Nat(F,F):'l'F(a) ~
~
~
= TT iEIe
a .• ~
4.
Thus define ~ E Mer (FX,FX) by
and define Ciy E Mer (FY ,FY) by
Then 'V f E Mer (X, Y) ,
=
=
E
F(t.
E
F(t. o )
ik, jl
ik, jl
=~
0
o
0
J-t-
J-t-
t~o
0
f
a,
0
F(t*J' o
J-t-
0
Ff.
Accordingly, the diagram
FX --------'> FX
FY --------'> FY
J
0
sik)
-t-
0
f
0
sik
0
sik~ )
5.
cortm1tes, rreaning that a. E Nat(F,F}.
And, by construction,
~. =
(li' so
1
IPF(a.) =
IT
a. .•
iE1C [Note:
g
x
1
The isorrorphisn IPF depends on the choice of the Xi.]
= g and
8.4
EXAMPLE
Take C'
8.5
EXAMPLE
Suppose that
let F
s: is a
= idc
(the identity functor) -- then
semisirnple nonoidal *-category -- then
g is a semisirnple *-category with
And X. 9 X.:::;
1
J
J:-.x., 1J-K
E9
kEI C
where
J:-. = dim M:>r (XL' X. 1J K 1
9 X.) ,
J
so Mer (X. 9 X. ,X. 9 X.) ::::: E9 MNJ: (e). 1 J 1 J k EIC .. _ 1J This said, let
- - -+ c. -
F = 9:C x C
Then
Nat (9,9) :::::
TT •
1,
]E'I
C
M:>r(9(X.,X.),9 (X.,X.» 1 J 1 J
6.
: : 1T . 'EI 1,)
Mor(X. ~ X. ,X. ~ X.) 1)
C
: : 1T
M
Ii:.
i , j , kEIe _
rf.. 1) Suppose that
~
"#.
1
)
(e) •
1)
0
is a semisinple *-category, let F:C
-+
FDHIIB be *-preserving
and put A
'F
= . eI 1E
B(FX.) 1
e
which, of course, can be embedded in
JT
B(FX )
i'Erc
i
(z
Nat(F,F».
Needless to say, .4y is a *-algebra, 1.IDital iff Ie is finite. Pi:.4y
'!he projections
i ) are finite d.irrensional irreducible *-representations.
-+ B(FX
any finite dimansional nondegenerate *-representatian of
t)..
Moreover,
is a direct sum of
finite di:rrensional irreducible *-representations and every finite dirrensional irreducible *-representation is 1.IDitarily equivalent to a Pi' Define now a *-preserving f1.IDctor
as in 8.2 -- then since
is an equivalence of categories iff F is faithful.
and F agree on JID:rphisms, it is clear that
faithful F faithful.
In fact,
7.
Assume therefore that F is faithful.
From the definitions, 'ITX . = p.]. (or still, ].
~:e -+
V
X ~ X
nX:X ~
V
EX:e
X
~
id
)
0
(idv
n*) X
0
(EX*
X -+ e
=>
-+
V
X.
And
(EX
~
Vx
X
~
nx ) == id Vx
==>
(id
XV
I.e. : (
~
V
~
id
XV
) == id
XV
•
The left duality ( x,Ex,n ) automatically leads to a right duality
* *) . XV ,nx,E x
x
2.
s: is syrrrr:etric
Now' assurre in addition that
(hence that the TX,y are unitary)
then the left dualiW (vX,E ,I1 ) gives rise to another right duality, viz. X X
v
(X ,EX
9.1
COHERENCY HYPOI'HESIS
'if
T , T x,vx x,vx
S:'
X E Db
E*
X
[Note:
0
= TO
.....
X, Vx
'rhe asyrrnetry is only apparent. -1 11 = T X X, Vx
0
o
'IX'
For E* X
E*
X
=>
In the
11* = EX X
0
T* vX,X
= EX
0
T
= EX
0
presence of 9.1, let
r
X = E*X
-1 vx,x
.
] T X,vX
0
I1 ). X
3.
thus
and
(x,r~,rX) is a left duality (x,r~,rX) is a right duality.
'lherefore
(r~ ~ idJ X
0
(id_ ~ rx) = id X X
'lhe relations
(id_ ~ r~) X
0
(r ~ id_) x X
=
id X
are called the conjugatE equations, the triple (x,rx,r ) be!ing a conjugatE for x. x N.B.
'I'J:e conjugatE equations imply that
4.
Having made these points, matters can 00 turned around.
£-
syrmetric strict nonoidal *-category
to each X E Ob
then
£
So start with a
has conjugates if one can assign
£ an object X and a norphism
such that the triple (x,rx,r ) satisfies the conjugate equations (here, of course, x
r
= T
X
0
X,X E.g.:
9.2
r )'
x
FDHILB has conjugates.
REMARK
If
£ has
conjugates, then
and right autoncm::ms (consider (x,rx,r x (rx ) *
in force:
= r X'
o
X,X
LEMMA
•
is left autonarrous (consider (x,rx,r » x
M:>reover, the coherency hypothesis is
while T
9.3
»'
£
Suppose that
£ has
r
X
=
T
X,X
conjugates •
Under the identification M:>r(X ~ Y,Z) ~ M:>r{Y,X ~ Z),
the arrows
are mutually inverse. •
Under the identification Mor(Y Q X,Z) ~
MDr(Y(z
Q
X) ,
5.
the arrows
are mu tually inverse.
V X E Ob
E.g. :
g, Mbr(X,X) ~ Mor(e,X ~ X).
9.4
LEMMA If
are conjugates for X, then r~ ~ id_
0
X'
rx
id_ ~ X
E
Mor{X,x')
is unitary. PROOF
Put
U = r~ ~ id_
0
X' = id
X'
~
r*X
id_ ~
X 0
r' ~ id ••• ). X X
Then the claim is that
-
UoU*=id
X'
U*
0
rx
U = id •
X
6.
And for this, it will be enough to consider U U
0
U*
= rx
~
= r*
~ id
id_ X,
X r'X X
U* ~
~
id_
X ~
Xl
G id_ X'
o (r'*
~
id
X
X
o
O
e
id
X,
= rx
U* ~ id )
0
r'X
~
(U*
0
U*
0
rx'
(id_ ~
0
o
= rx
So write
id ~
o
X'
X
U*.
0
0
id
X,
~
~
rX
id X
e X'
)
id
X'
o;d n rX - wn ~'d (r '* n;d w • ~ w _ X X X, X e Xl o
= rx
~
o (r ' * X
0
'd) X ~ X'
~
id
X'
id_
X,
e
id ~ id X X ~ X' o
0
~
rx
X ~
Xl
id
X'
id
Xl
o r'* X
~
id
X
~
id
~ id
_) X e X,
7.
= r'*
~
X
=
o
id
~
X' ~ X
~ rX ~ id
id
X
X'
= r'* X
id
o
id
Xl
Xl
r* ~ id X XI
rXl * ~ idXl 0
o
~
rX _
~
o
id
~
id_
XI X' ~ rl
id
0
XI
X'
id
XI o
id XI
o
= r
l
*~
X
id
X'
o
id XI
= r'*
X
~
X
id
Xl
id
Xl
~ rX ~ id
_)
X ~ Xl
8.
o
= r'* X
= r'*
(id
X,
~
~
X
= r'* X
id
o
id
X,
id
X'
X,
id
0
id
0
X'
X
= r'*
o
~
X'
~ i~ ~
X'
X,
Q id
(id
0
X' ~ X'
o
Q
X
~ X'
o
id
X'
X'
id
id
id
id
X'
r' X
= id
X'
[Note:
Evidently, r' = X
Conjugates are therefore detennined up to "unitary equivalence". Put
9.
~
E MJr(X,X)
is unitary and it can be verified. by computation that the assignrrent X defines a twist
!...EMMA
PRCX:>F
V
~
This fact, however, is a trivial consequence of the following
Q.
result.
9.5
+
X E Ob
g,
We have
On the other hand, there is a conmutative diagram T
e,X
X=e~X
---------:>
X
~
e
=X
X~X~X
----------~>
X
~
X
~
T
X
~
X,X
so
=
rx
~ i~
rx
~ i~
0
T_
X
0
~
X,X
And 0
T_
X,X
~ i~
rX ~ i~
X,
10.
[Note:
= r*x
0
T
= r*X
0
T
x,xx,X
~
in
0
T_
T
0
-x
X,X
~ i~
~
in
0
i~
-x
X,X
Therefore, in the tenninology of §6, C is an even ribbon category.]
9.6 REMARK V f E M::>r(X,X) , the diagram T
X,X
X~X
id ~ f X
;>X~X
1£9
1 x~x
ccmru.tes.
T
;>x~x
x,x
'Iherefore
r*X
0
f ~ id
X
= rx = r*
X
0
T
0
rx
0
f ~
x,x 0
id ~ f X
0
id X
rx.
0
T
x,x
0
rx
id X
11.
g has conjugates, recall that g is left
Maintaining the supposition that
autonom:ms with left duality (x,rx,r ) (cf. 9.2), thus by definition the catex gorical dirrension of X is the arrow
-
T
rX
r*
X,X X > X QX--->XeX---i>e
e
(cf. §6).
But
S)
the categorical dirrension of X is the carposition
r*
T
0
X
T
o
X,X
= rx ~
[Note:
Since Q = id,
V
f
E
o
X,X 0
dim
r
X
rx E Mer(e/e)
x.
Mer (X,X) , o
= r*
0
X
= r*X
0
= r*
0
X
N.B.
U:X
-+
XI
T X,X
0
id X
~
T
0
X
(f
id
X
~
for
X
id )
o T
X
o
X,X
~
-r X X,X
r
X
T o r X X,X
(cf. 9.6).]
dim X does not depend on the choice of a conjugate for X. is unitary, then
o
Indeed, if
12.
= r*X
9. 7
J:».1MA
0
If
then
is a conjugate for X
~
Y, where
[The proof that
id
YBX
will be left to the reader but
~
shall provide the verification that
r-X
~
Y
= T o rX Y B X,x 9 Y
0
rX 9 Y
'!hus write T
Y ~ X,X
~ Y
6)
Y·
13.
=
TOT
X,X ~ Y ~ 'l
T
0
Y,X
~
= id
X
~
Y
~
X
Y, Y
= id
~ T
id
~ T
X~ X
id
id
=
id
(r
X
~
~
~
~ T
X~X
Y,Y
)
o r
~
X,X ~ Y ~
Y
0
~ i~
0
r
Y
Y
o ide ~ ry
id_ Y~Y
~
ry o rx
X~X
o r o r = id_ Y X
X~X
Therefore
=T
(cf. 4.3).
'l,e
id_
o id_ Y,Y
T
0
Y
Y~Y
o rX
~ T
Y
X
id_
Y,Y
XSX
~
rX ~ Y
0
o rX
~ T
rX
0
Y
Y,Y
X~X
=
~
Y,X
0
Y,Y
X~X
=
rX
~ TOT
X
=
Y,X ~ X ~ Y
id 9 X~ X
o
r
r
Y
X
0
rX
~ ry o r x •
14.
For all X,Y E Ob
g,
the map Mor{X,Y)
v
that sends f to
N.B.
-+
Mor{Y,X)
f is a linear bijection.
Here, as will Vf
re
recalled fran §5,
= r*
~ id
X
Y
id
0
Y
~ f ~ id
X
NCIW" put
thus
f+
= id ~ Y
r* X
0
id
Y
~ f* ~ id
and
- - • f + E Mor{X,Y) Pro]?8rties: 1.
2. 3.
9.8
(f
LEMMA
0
g) +
= f+o g+.
Given f E Mor{X,Y), we have
X
15.
PROOF
Start wi th the IRS and
= id
y
~ r* ~ in
-x
x
0
write
id_ ~ f* ~ id_ ~ in
y
0
-x
x
ry ~ id_ ~ in
X-X
But o
r
Q
id
X
=>
= id_y
~
in
-x
~
rx
0
id
y
~
f*
id
Q
e
ry
0
=>
= id_y
~
rx*
~ in
-x
y
~ in . ~ r
-x
x
0
id ~ f* y o
ry
0
r
y
0
r • x
16.
= id
9 f*
ry .
0
y
9.9
REMARK
MJr (X, Y) satisfies the equation
Suppose that T E
Proof: f+
= id ~ r* Y X
0
id
= id
0
id ~ f*
Y
en
~ r*
Y
X
~
f*
~
id
0
ry ~ id_
X id
0
X
y
X
the other hand, T= To id X
= T
0
= T
~
id ~ r* X X ide
rx ~ id_ X
id_ ~ r~
0
0
~
rx
id_ X
X
= id_y
~ rx
= id-
~
= id_y
~ rx
Y
0
0
T
~
id
rx
0
~
id_
X~X
r*
X
0
T
~ i~ ~
X
id_
o
rx
~
0
T.6;O
i~
0
rX
~
id_ X
X
id_ X
0
id..X
17.
[Note:
It is thus a corollary that if id
G f*
ry = 0,
0
y
then f +
= 0,
so v
a => (f )**
9.10
SCHOLIUM
v
= 0 => f
= 0.]
f+ is the unique elerrent of 1-br (X, Y) such that f+ G id
.-X
[Note:
= 0 => f
0
r
= idY G f*
X
0
r . Y
v f is the unique elerrent of IVIor (y,X) such that
so f+ is the unique elerrent of Mor(X,Y) such that
r* y
9.11
LEMMA
0
id
-y
G f+
= r* X
f* G id_o]
0
X
Sup];X)!'3e that
F :C
is symretric and unitary °
Given X E Ob
r
=
FX
(~
X,X
FDHIIB
-+
£,
)-1
0
put Frx
0
~
r-FX = T o rFX o FX,FX Then the triple (FX,rFX,r
FX
) is a conjugate for FX.
18.
PR(X)F
What we know is that
hence
F(id_
e
r~)
= id _, x e id) X FX
F(r
0
X
and what we want to prove
that
(id _
e rFX* )
(r
0
FX
FX-
!
= id
.
FrX
sl':
id _)
FX
FX
The UIS of the first of these is the canposition o Fr~
~
0
X,X o
o T
FX,FX
F being unitary.
) -1
(:::: _
0
X,X
0
..... (l
_ ~ 1~,
Write o
T
FX,FX
(:e: ) -1 X,X
0
Frx
0
i;
~ i~
=
= T _
FX,FX
~ i~
0
(::...
X,X
)
-1
~ i~
0
Frx @ i~
0
i;
~ i~.
19.
Taking into aca:Junt the corrmutative diagrams
e == e
~
FX - - - - - FX
Fe ~ FX - - - - - > F(e ~ X)
,X
Fe ~ FX - - - - - - - - - - ' > F (e ~ X)
FrX
!!
~1
F
(X
~ X) ~ FX -
1
> F (X ~ X ~ X) , X
'fNe
have
This leaves
Next
F (rx II
~
x,x
lllx)
20.
Since F is syrmetric, there is a corrrnutative diagram top
-
-
F(X 9 X ~ X)
----------------->
FX~FX~FX
1
F{TX,X II i,\l
FX ~ FX ~ FX -------------------:> F (X ~
X~
X) •
bttm Here "top" is the COI'!'IfX)sitian
=_
x,x
~ i~
FX ~ FX ~ FX --------------------,> F
(x ~ X)
~ FX
X ~ x,x ----------------~> F(X ~ X ~ X)
and "bttm" is the composition
= _ ~ i~ X,X
FX ~ FX ~ FX - - - - - - - - - - > F (X ~
X ~ x,X ----------------...,> F (X ~
X)
~ FX
X~
X) •
Therefore o
X
~
X,X
=x,x_!
i~
o
T _
!
i~
FX,FX
=> T _
FX,FX
~ i~
0
(=_ X,X
)-1 ~ i~
0
(=_ X
) ~
X,X
-1
21.
=>
Analogously,
~ ~-l
id
-:FX-
i~
0
0
Fr*
0
H
XX,X
F(i~ ~ r~)
0
0
H
X,X 9 X
i~ ~ E_
X,X
•
So, in sumna:ry,
o E
_
0
i~ ~
E_
X,X 9 X
0
X,X
(E
_) -1
X,X
thus tn finish, it need only be shown that
= id
F(X ~
X~
X)
'lhis, however, follOtN'S from the carrmutative diagram
~ i~
0
(E
)-1
X 9 X,X
22.
-
-
FX~FX~FX
-
F(X ~
x
~
9.12
REMARK
~
-
FX ~ F(X ~ X)
FX
1x,x
1
X,X F (X
X)
FX~FX~FX
E
~ X~
X)
============
F(X ~
111 X
x ~ X).
We have
-
r
(E )-1 FX = X,X
0
FrX
0
~.
In fact, the RHS eq:ua1s
(E
_)-1 X,X
0
FT
0
X,X
Fr
x
0
~
and there is a carmutative diagram
X,X
FX ~ FX - - - - - - - - - - - >
F
(x ~ X) FT
T
X,X
FX,FX FX
~
FX - - - - - - - - - - - > F (X X,X
~
X).
1.
§10.
let then
s::
s::
TANNAKIAN CATEGORIES
be a sym:retric strict ITDnoidal *-category which is essentially small -
is said to be tannakian. if the following conditions are net:
dim M:>r (X, Y)
bjects, direct sums, and conjugates.
T3:
s::
has a zero object.
T : 4
e is irreducible.
10.1
RE'MARK
10.2
EXAMPLE
A tannakian category is necessarily semisimple, hence is abelian.
let CP'IGRl? be the category whose objects are the carpact
Hausdorff topological groups (in brief, the "conpact groups") and whose ITDrphisms are the continuous ham::m::>rphisms.
Given an object G in this category, let Rep G
be the category whose objects are the finite d.inensional continuous mitary repre-
senta tions of G and whose ITDrphisms are the intertwining operators -
G
then
is tarmakian (define r and r by rA = ALe. i
~
~
e.
~
(A E C rA = ALe. . ~ ~ ~
where {e.} ~
c
H
(=
e)),
e.
~
an orthonorm.al basis for the representation space and {e.} cHis ~
2.
its conjugate). [Note:
In particular:
FDHIIB is tannakian (take G
=
{*}).
The construct Rep G is annestic and transportable, so we can and will
ass1..lllE that its nonoidal structure is strict (cf. 3.12).]
10.3
An additive functor F:A
RAPPEL
+
B between abelian categories A and
B is exact if it preserves finite limits and finite colimits.
Accordingly, since a tarmaJdan category is not only abelian but also autonorrous, V X E Ob
g,
the functors -
X
§II
X, lham(X,-)
§II - ,
rham(X,-)
are exact. If C is tannaJdan, then e is irreducible and dim:Ob
g+
MDr (e,e)
has the following properties. l.
dim X
= dim
2.
dim(X
§II
Y)
=
3.
dim(X
e
Y)
= dim
X + dim Y.
4.
dim e
= 1,
dim 0
= O.
10.4 PRCX)F
LEMMA
X. (dim X) (dim Y) •
If X is not a zero object, then dim X (= rx
0
rx)
First, from the positivity of the involution, dim X > O.
contains e as a direct sumnand, thus (dim X) 2
~
1 =>
dim
X
~ 1.
~
1.
But X
§II
X
3.
If dim X = 1, then X
[Note:
Given X
;J!
0 in Ob
g,
Q
X ~ e ::::
XQ
X.]
define
as in 4.5.
N.B.
~ is a homJrrorphism from Sn to
the unitary group of Mor(X,X) •
Put
and for n E
~,
.po =
e,
X SYID:
= n! -1
put
n
Then
S~
are projections.
L cC;
O'cpn
-X
Hn-- (0')
4.
10.5
lEMMA We have
=4 n. PR.CX)F
(dim X) (dim X - 1) ••• (dim X - n + 1).
The key preliminary is the observation that tr~ (~(O»
= (dim X) #0,
where #0 is the ntnnber of cycles into which 0 decarrposes, thus
Bu t for every complex ntnnber z,
L
(sgn O)z#O
=
z(z - 1) ••• (z - n + 1).
0E$
n
10.6
THEOREM
'if
nonzero X in Ob
~,
dim X E N. PR.CX)F
Let An (X) be the sul::x:>bject of
i~try f:An (X)
-+
~ such that f
corresponding to Al t~.
= Al t X -n
f*
0
~
then
tr~(Alt~)
= tr
~
= trA
(f
0
(X) (f*
n·
f*)
0
f)
(cf. 6.3)
Fix an
5.
= trA
n
(X) (idA (X» n
= dim An (X) ~ 1
en
(cf. 10.4).
the other hand, thanks to 10.5,
tr~
(Alt~)
is negative for same n E N unless dim X E N.
10. 7
LEM-1A
Let d
dim X -- then
The lsorrorphism class of Ad (X) is called the detenninant of X (written det (X».
Properties:
1.
det (X) ::::: det (X) ;
2.
det(X E9 Y) ::::: det(X)
3.
det (X ED
X) ;:
e.
~
det(Y) i
1.
§ll.
FIBER FUNCTORS
let C re a tarmaldan category -- then a syrmetric embedding functor f: C
-+
FDHILB
is called a fiber functor. E.g.:
Take
£ = Rep
G (cf. 10.2) -
then the forgetful functor
U: Rep G
-+
FDHILB
is a firer functor. N.B.
It is a nontrivial result that every tannakian category admits a fiber
functor (proof omitted) •
11.1
REMARK
let f :C
re a fiber func tor •
-+
FDHILB
Consider A~ ;1
= e
B(fX.),
~
iEIC
vie\lied as a subset of Nat(f ,f) -- then the coinverse is the map S:A f
-+
Af
defined
by
matters reing slightly imprecise in that the identification f (X ~
has
reen
suppressed.
X~
X) :: fX ~
r.X
~ fX
It is not difficult to see that the equation defining S (a)x
is independent of the choice (x,rx,r ) of a conjugate for X and V f E MJr(X,Y), x
2.
the diagram
cormutes.
~~------------~>
fX
----------------~>
fY
Algebraically, S is linear and antimultiplicative. S
0
*
0
S
0
*
Moreover,
= idA '
F
hence S is invertible. [Note: detailed.
There are various relations arrong lI,£,S which, hoI;..iever, need not be Still, despite appearances, in general (AF,lI,£,S) is not a Hopf
*-algebra but rather in the jargon of the trade is a "COCOImnltative discrete algebraic quantum group".]
Write ff (£) for the full subcategory of
"Whos:= objects are the fiber functors -- then ff(9 is a groupoid (cf. 5.12).
11.2
THEOREM
ff(£} is a transitive groupoid, i.e., if F ,F
l
2 are fiber
functors, then Fl' F2 are isarrorphic.
~fini tian:
fonnation a: F1
-+
Given fiber functors F1 f F2 f a
uni~
rronoidal natural trans-
F2 is a rronoidal natural transfor.mation such that V X E Ob
£,
3.
is illlitary . W'ri te ff* (9 for the category whoa: objects are the filler filllctors and whoEE norphisms are the illlitary rronoidal natural transfonnations -- then ff* (£)
is a subcategory of ff (£).
11.3
THEOREM'.
ff*(9 is a transitive gro'l.J±X)id, i.e., if Fl ,F2 are filler
fimctors, then Fl'F2 are illlitarily iSOlIDrphic.
Obviously, 11.3 => 11.2. As for the pr.]
'lb fix notation, bear in mind that there are isarn::>rphisms
subject to the compatibility conditions enl..llterated in §2. Let AO (FI' F2) be the complex vector space
e
Mor(F X,F X}.
X E Ob C
that is
Fl (X ~ Y).
5.
PRCX)F
let
1A
o
Then lA
o
is the mit.
E.g.:
= [e,,,,c-l
0
(c-'" 2 )-1]O.
Consider
l [X,'I'A..] O· [Ce,,,,
0
(C-'"2 ) -1] 0
=
[X, u] 0'
the claim being that the cornposi te (~2
X,e
)-1
0;:;1
-X,e
reduces to ¢ itself.
'Ib see this, recall that the composition
is the identity rrorphisn of fIX and the composition
is the identi ty rro:rphism of f 2X.
NCM write
6.
"d_ x = ~-r
0
("d ~ f
x
0
¢
0
"d_ X! ( ~) -1 -1
~-F
0
(1 ~
0
112
where
let
be the projection, and put
[x,¢]
11. 6
EXAHI?IE
11. 7EXAMl?IE
= pr{x,¢]O.
let f:X -+ X be an isarorphism -- then
let
(2 -1 ~)
0
~
2)
0
idf X 2
7.
Then
[NOte:
11.8
We also have
REMlillK
choice of X and
f X 2
~
f Y 2
1~ ~ w
.
9.
'!hus
_1
~,X
But
,I, n
~
't'
't'
0
~
(~2 0
~Y,X
)-1
there are also comnutative diagrams ~l ~X,Y
F1X ~ FlY
T"lX,flY
- - - - - - - - : : > 1'1 (X ~ Y)
1
FlY ~ F1X - - - - - - - - > 1'1 (Y
~ X)
~l ~Y,X
and
F2Y ~ F2X - - - - - - - - > 1'2 (Y
T"2y ,f2X F2X
1 @F2Y - - - - - - - - - ' > ;::;,2 ~X,Y
'!hus
~ X)
1'2 (X ~ Y) •
10.
Let f
= TX,y
and put
a
1 = -~X,y
0
~ Q_ ,I. ~
~
0
_2 -1 (M-X,y )
0
f E M:>r(X Q Y,Y g X)
and
M:>reover
~anwhi1e
= id;: so
Therefore
=>
2
(X Q Y) ,
~ T )2 y,X·
11.
Given [X,$]O' choose a conjugate (x,rx,rx ) for X and let
where
¢
is the ccroposi tion t,;1
~ id
~~-------:> FIe ~ F 2
w: } ~ -1
x
id
X,X
id ~ $* ~ id - - - - - - - - - - ' : > F IX
~ F 2X ~ F 2x
id ~ 22
X,X - - - - - - - > FIX ~ F2 (X ~
id ~ (t,;2)-1 - - - - - - - - ' 7 > FIX ~ ~ N.B.
We have
X}
FIX'
12.
and
(cf. 9.11).
'Iherefore
[Note:
§5,
or still,
'Iherefore
Therefore the image of [X,] conjugate for X.
0 in
A(f ,f ) is independent of the choice of a l 2
13.
A(fl,f2) is a conmu ta tive unital *-a1gebra.
Sumnary:
Accordingly, to coop1ete Step 1, it remains to construct an iSOItOrphism
en
Bu t
general grounds,
the pairing
tha t ::ends t/l
en
x
I/J to tr (t/l
0
I/J) is nondegenera te, thus
the other hand, Nat(f ,f ) consists of those e1erre.'1ts 1 2
9..lch that
V f
E Mor(X,Y),
that vanish identically an 10 (fl'f2) •
'lb
characterize the latter, take an
14.
= 0 or still,
for all
I.e. :
From the nandegeneracy of the trace, it then foll()\,\lS that
iroplying thereby that
11.12
LEMMA
Under the bijection
the nnnoidal natural transfonuations correspond to the nonzero multiplicative
linear ftm.ctionals on ACFl' f 2) • PROOF
To say that a linear functional on A(fl'f ) corresponding to an 2
a E Nat (fl' f 2) is mul tiplicative a;rrpunts to saying that
15.
=
•
for all
Since < -,a> is null on 10 (Tl'T2) , it suffices to work upstairs, hence explicated 'Ne
have
trTl (X 0 Y) ( ~l ~X,y
= trT
(~2
n 11,
0
~ ~ ~
X 0 T Y «~ 1 - 1
0
0
-X,Y
)-1
~) ~ (~
Therefore ~
0 Y=
~2 ~X,y
n
0
~ ~
ay
0
(~l )-1 ~X,Y ,
the condition that a be! rronoidal.
[Note:
Tacitly,
= 1
o
or still, < [e,t,;
1
2 -1
0
(t,;)
lO,a> = 1
or still,
from which the conmutativity of the diagram
0
0
) ~ 0 Y
ay))
16.
a
e > F2e
FIe
Fl (X ~ X)
"x !!
ax
1
1ax - - - - - - > F2 (X ~ ;::;2
Q
X
X)
X,X \I.e
see that
and from the carrrmJ.tative diagram a X~X
- - - - - - - - - > F (X ~ 2
X)
1J'i~
\I.e
see that
Recalling now that
(cf. 9.12)
19.
we have
=
(t:?) -1
=
o
r*
f2 X
(t.?) -1
;::;2
0
H:?
-) X,X
0
X,X
0
r*
f2 X
0
a
-1
o
a
0
;::;1
X9X
0
X,X
_1
.:::.
X:9X
X,X
X,X
-- r*f X' 1
as c1airred.
'lhe results embodied in 11.12 and 11.13 finish Step 2 of the program, which
leaves Step 3 to be dealt with. Put Af
f
l' 2
11.14
LEMMA
=
$
iEIC
Mor(f2x.,f1 X.). 1 1
'lhe linear map
that sends
cp.1 E Mor(f ,fIX') 1 2X. 1 to [X., cp .] is an isarrprphism of vector spaces. 1
1
20.
PRCX)F Every A E ACF'1,f'2) is an [X,q,] (cL 11.8) and every [X,q,] is a of elerrents [X.,q,.] with X. irreducible. J.
J.
Therefore q is surjective.
J.
That q is
injective is a consaquence of the fact that i ;.e j => !!br(X. ,X.) J. J
=
{Ole
Put
Then trere is a direct sum decornposi tion
A(f'1,f'2)
= e A .• iEIC
J.
Define a linear functional
by taking it to be zero on A. if i dres not correspond to e but on A , let e
J.
w([e,q,])
=
1 -1
(~)
q,
0
0
~
2
E
C.
11.15 LEMMA V A ;.e 0, w(A*A) > o. PRCX)F Write A = E [X., q, . ] , • J.
J.
J.
where the Xi are irreducible and distinct -- then i;.e j => w([X.,q,.]* • [X.,q,.]) = O. J.
In
J.
J
]
fact, l1:Jr(e,X. ® X.) ::::: !>br(X. ,X.} = {O} J. J J. J
SlID1
(cf. 9.3),
21.
S)
e is not a subobject of
wi th X irreducible.
~ X.. ]
X.].
One can therefore assun:e that A
= [x, cp]
Recall now that
This said, let r p Then
pX
= X
X
0
r* X
= PX and
r =
X
0
r*
~
X
= Px·
I.e. :
is a projection.
write A*A = [X,cp] * • [X,cp]
= [X,(p] •
[X,cp]
o 31 X,X
o;p ~
cp
0
G 2 ) -1] X,X
;t
0
22.
~ ~ n 'I' ~ o 'I'
(~2_ _
0
X,X
=
=
[X
_1
~ X , F1 (PX)
0
~
(* r
FIX
~
o
o
1> ~ 1>
n
~
0
X,X
O:~: ) -1]
0
X,X
1> ~ 1>
0
X,X
'I' ~ 'I'
0
X,X
m:
1 [e,F r 0 1 X ~
= -1
M
~
)-1]
(3:
)
-1
o
F r ]
X,X
r F X )[e,,,,~1
2 X
(cf. 11.7)
(~2)-1] '"
0
2
where
Then
when viewed as a constant, is normegative. the unique element of
;t
0 =>
;t
~ id_ X
-T 2
0
r~ T
2X
= id
F
g 1>*
0
r~ X
1
X-
= 0 => 1> = 0 => (
v
1»*
O.
Proof:
¢ is
Mor(F2X,F1X) such that
¢ so
But 1>
= 0 => (
v
1»** = 0
=>
V
1> = 0 => 1> = O.
'1
(cf. 9.10),
23.
[Note:
'lb justify the equation
write
Then
= w(A*B). Then
equips AtF ,f ) with the structure of a pre-Hilbert space w.r.t. l 2
which the left mu1 tiplication operators
are continuous.
Denoting by H(fI' f 2) the Hilbert space corcpletion of A(fl' f 2) ,
it thus follows that A(f ,f ) admits a faithful *-representation l 2
hence A(fl'f2 ) admits a C*-no:nn as clain:ed in Step 3.
1.
THE INTRINSIC GROUP
§12.
let g be a tannakian category and suppose that f:C -+- FDHILJ3
is a fiber flIDctor -- then its intrinsic group G is the group of m.itary nonoidal f natural transfornations a:f
-+-
f, Le., in the notation of §ll,
where M:>r(f,f) is COIl'pUted in ff*(g). So G
f
c
Tf
U(fX),
XEOb C
U(fX) the compact group of lIDitary operators fX
rr
-+-
fX.
And G
f
is closed if
U(fX)
XEOb C
is equipped with the prcx1uct topology, thus G is a compact group. f N.B. Define
12.1 :£ -+-
LEMMA
3 a faithful synmetric rronoidal *-preserving flIDctor
Rep G such that u f
is the forgetful functor.
0
= f,
where
2.
pRCX)F IJe!fine
~
and on no:rphisms f:X entities per f.
'Ib
on objects by
-+
Y by
~f =
ff (cf. 8.2) and take for 6;., E the corresponding
see that this makes sense for E say, one nrust check that
EX, Y is a no:rphism in Rep Gf , viz.:
But this is obvious since the diagram M
-X,Y --------;> f (X
~ Y)
l"x - - - - - - > f (X
II Y
~ Y)
-X,Y conm.utes.
'!hat
~
I-bre is true:
is syrmetric is equally clear.
~
is an equivalence of categories.
ren:ains to establish that
12.2
REMARK
~
is full and has
Ii
Because
~
is faithful, it
representative image (details below).
The category Repfd Af is a semisimple synmetric nonoidal
*-category which can be shown to have conjugates, thus RePfd Af is "alnost" tannakian.
Specializing 8.14, it was pointed out in 8.16 that the
there is a syrmetric nonoidal equivalence
£
-+
Repfd Af"
"~n
defined
Denote now by Repfd G
f
the category whose objects are the finite d.lm;msional continuous representations
3.
of G and whose norphisms are the intertwining opera tors -- then the inclusion f functor
en
is an equivalence.
the other hand, there is a canonical functor
and it too is an equivalence (a nontrivial fact) .
12.3
LEMMA
If X E db
S is
irreducible, then the complex linear span of
(a E G ) is dens:! in B (f'X) •
the 'TTX (a)
f
12.4
LEMMA
If X,Y E db
S are
irreducible and noniso.rrorphic, then the complex
linear span of the 'TTX Ca) Ea 'TTy (a) (a E G ) is dense in B (f'X) Ea B (fY) • f
12.5
REMARK
If Xl, .•. ,Xn are distinctelenents of Ie' then the complex
linear span of the
is dense in
'lb
prove that 4'> is full,
'WI9
shall appeal to 7. 9 .
(a) X irreducible => 4'>X irreducible.
In fact, thanks to 12.3, the only
T E B (fX) that intertwine the 'TTX (a) (a E G ) are the scalar rnul tiples of the identity. f
(b) X,Y irreducible and noniso.rrorphic => X,Y irreducible and nonisarrorphic.
4.
But then Tu
= vT
for all u E B(fX), v E B(fY) (cf. 12.4).
to conclude that T
= 0,
Now
take u = 0, v
=1
hence H TIl
T1
iT > H , TI2
H
TI2 TI2 (a)
thus the string
defines an elerrent
cd a) E MJr (D , D) ,
6.
where technically
12.8
LEMMA
r is a continuous injective harroIrorphisn.
[This is imTedia te from the definitions.]
In fact, r is surjective, hence G and
% are
isorrorphic.
[If r \'Nere not surjective, replace G by rG and think of G as a proper closed subgroup of
%-
then there 'WOuld l::e an irreducible representation of
contains a nonzero vector invariant under G but not under
%.
% that
This, however, is
i.n:possible:
is bi jective.]
12.9
THEOREM
Up to isonnrphism in CPTGRP, G is the
II
intrinsic group" of
Rep G. [If F : Rep G
r :: %
is a fiber functor, then G
12.10
-+
FDHILB
(cf. 12.7).]
REMARK Compact groups G,G' are said to l::e isocategorical if Rep G,
Rep G' are equivalent as nnnoidal categories.
In general, this does not rrean
that Rep G,Rep G' are equivalent as sy:rrrretric nnnoidal categories and G,G' may very \'Nell l::e is:::lCategorical but not isonnrphic.
1.
§13.
CLASSICAL THEORY
A character of a carrmutative lUlital C*-algebra A is a nonzero haroIrorpbism
w:A
-+
C of algebras.
The est of all characters of A is called the structure space
of A and is denoted by b. (A) • N.B.
We have
ret w E b.(A) -
b.(A)
=~
(A
b. (A)
~ ~
(A ~ {O}).
= {O})
then w is necessarily oolUlded.
13.1
LEMMA
N.B.
The elen:ents of b.(A) are the pure states of A, hence, in particular,
are *-horcorrorpbisms:
\if A E
In fact,
A, w(A*) = w(A) •
Given A E AI define A:b.(A) -+ C
by A(w) = w(A).
Equip b.(A) with the initial topology determined by the A, i.e., equip b,(A) with
the relativised weak* topology.
13.2
LEMMA
b.(A) is a compact Hausdorff space.
2.
If X is a compact Hausdorff space, then C (X) equipped with the supremum nonn
II f II
= sup
xEX
If
(x)
I
and involution
= f(x)
f* (x) is a camu.:ttative unital c*-algebra.
ox
MJreover, V x E X, the Dirac neasure
E il(C(X)) and the arrow X + Ll(C(X))
x
+
0
x
is a horrearorphism.
13.3
LEMMA A E C(Ll(A»
and the arrow A+C(il(A))
A+A is a unital *-ison:orphien.
N.B.
If A = {a}, then Ll(A)
the empty function
Notation:
(~= ~
=~
and there is exactly one map
~
+ C, nanely
x C), which we shall take to be O.
ret CPTSP be the category whose objects are the carpact Hausdorff
spaces and whose :m;)rphisms are the continuous functions. Notation:
ret cc.MUNC*AIG be the category whose objects are the cormutative
unital C*-algebras and whose norphisrns are the unital *-hDm::mJrphisrns. let X and Y be compact Hausdorff spaces. function -- then
C(1".(A»
i···
•1 B
;>
C(1".(B».
id :::: 1".
o
C
id :::: C
o
1"..
:::'B Therefore
The category CPTSP has finite products with final object {*}. category CCMUNC*AlG has finite coproducts with initial object C.
Therefore the 'lb explicate
the latter, invoke the nuclearity of the objects of CXMUNC*AlG, thus
A ~max B = A ~. B, nun call it A ~ B -- then
and there are arrows A+A~B
13.5
EXAMPLE
We have
C({*})
~
C and
C(X x Y)
~
C(X)
@C(Y)
5.
13.6
REMARK
let A be a COImn.ltative l.IDita1 C*-a1gebra -- then the algebraic
tenror proo.uct A Q A can be vie\'Iled as an invo1utive suba1gebra of A ~ A. point is this: A
Q
Since A
A __ m_>A with meA
~
Q
Another
A is the coprcxluct, there is a canonical arrow
B) = AB, i.e., the restriction of m to A
Q
A is the
Iml1 tip1ica tion in A. [No te :
If A , A2 ' B are carrmu ta tive l.IDi tal C*-a1gebras and 1
are l.IDital *-horronorphisms, then the diagram A1 ----i> A1 ~ A2 < - - A2
~1
1
1~2
B
B
admi ts a l.IDique filler
such that
13.7
RAPPEL
let
then a group object in
g
be a catego:ry with finite proo.ucts and final object T --
g consists ~:G
x G
of an object G and rrorphisms +
G, n:T
such that the following diagrams COImn.lte:
+
G, l:G
+
G
6.
11 x id G G x G x G
> G x G
1~
1
id x 11 G
G x G
>
G,
11
G x T pr
n
n
i'\; x
T x G
> G x G
1
1
l
pr 2
11
G,
G
1
11
G
G,
G -----------> T
1
{l,idG}
G x G --------
> G x G
1
G ----------~> T
(idG,l)
x id G
G,
1
G x G --------
G.
11 There are obvious definitions of internal group harron:orphism G of internal group harron:orphis:ns G harron:orphism idG:G
-+
G.
-+
G', G'
-+
G' "
-+
G', composition
and the identity internal group
Accordingly, there is a category GRP (~) whose objects are
the group objects in ~ and whose nvrphisrns are the internal group horrorrorphisrns.
[Note: I, then
'We
If instead
~
is a category with finite coproducts and initial object
put
and call the objects the cogroup objects in group honarorphisms.]
13.8
EXAMPLE
Take C
= SET
-
then
~
and the norphisrns the internal co-
7.
13. 9
ID'IrvlA
'V'e have GRP (CPTSP)
13.10
REl'1A...~
=
CP'IGRP.
Too forgetful func tor CPTGRP -;.- SET
has a left adj:>int.
Given a s:t X, equip it wit..l1 the dis:!rete topology,
Proof:
form the aSSJCia tEd free topological group F
gr
(X), and consider its Ebhr CC>I:r{')act-
ifica tion.
A corrmuta live Hopf C*-algebra is corrmuta tive unital C*-algebra H together with unital *-l:'om::>Irorphisns
A:H -;.- H @H, E:H -;.-
C, S:H -;.- H
for which t:l"e following diagrams corrmute: A
H 6
>H~H
1~ ~
1
H~H
> H
~
H
~
6
H,
A ~ i~
H
I-I
6
1 > H ~~E
~
H
1
1
in l
6
C,
H~H
1
H~H
H
ll2
> C ~ H,
E ~ i~
8.
C
H< (~,S)
I
1
(S/i~)
E
H,
H0H
