reconstruction theory - Department of Mathematics - University of

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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

~



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)



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*



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