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Diagrammatic Algebra

Aaron Lauda

Columbia University

December 8th, 2008

Available athttp://www.math.columbia.edu/ lauda/talks/diagram.pdf

Aaron Lauda (Columbia University) Diagrammatic Algebra December 8th, 2008 1 / 42
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Plan

Diagrammatic calculus has a broad range of application

Higher-categories provide a unifying framework for understanding

various diagrammatic calculus that appear in mathematics.

We will look at application of this diagrammatic framework bylooking at relationships between biadjoints in 2-categories,

Frobenius algebras, and low-dimensional topology

We will also look at an application where this graphical calculus

shows up while categorifying quantum groups.

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Plan








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Plan








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Plan








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CategoriesA category consists of

a collection of of objectsx,y,z,. . .

a set of 1-morphisms y xf

composition of morphisms x yg

zf = y x

gf

such that

composition is associative: given Z yh

xg

wf

wehave

(h g) f =h (g f)

identity morphisms: for each objectxa morphism x x1x such

that

y xf x

1x = y xf = y y

1y x

f

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zf = y x

gf

such that


xg

wf

wehave

(h g) f =h (g f)


that

y xf x

1x = y xf = y y

1y x

f

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zf = y x

gf

such that


xg

wf

wehave

(h g) f =h (g f)


that

y xf x

1x = y xf = y y

1y x

f

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zf = y x

gf

such that


xg

wf

wehave

(h g) f =h (g f)


that

y xf x

1x = y xf = y y

1y x

f

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zf = y x

gf

such that


xg

wf

wehave

(h g) f =h (g f)


that

y xf x

1x = y xf = y y

1y x

f



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zf = y x

gf

such that


xg

wf

wehave

(h g) f =h (g f)


that

y xf x

1x = y xf = y y

1y x

f

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zf = y x

gf

such that


xg

wf

wehave

(h g) f =h (g f)


that

y xf x

1x = y xf = y y

1y x

f

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zf = y x

gf

such that


xg

wf

wehave

(h g) f =h (g f)


that

y xf x

1x = y xf = y y

1y x

f

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2-categoriesAstrict 2-categoryis given by

objectsx,y,z,. . .

morphisms y xf

2-morphisms between 1-morphisms xy

f

g

composition for 1-morphisms

horizontal composition xy

f

g

yz

f

g

= xz

ff

gg

vertical composition xy

f

g

= xy

f

g

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objectsx,y,z,. . .

morphisms y xf


f

g



f

g

yz

f

g

= xz

ff

gg


f

g

= xy

f

g


i
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objectsx,y,z,. . .

morphisms y xf


f

g



f

g

yz

f

g

= xz

ff

gg


f

g

= xy

f

g


2 i
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objectsx,y,z,. . .

morphisms y xf


f

g



f

g

yz

f

g

= xz

ff

gg


f

g

= xy

f

g


2 t i
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objectsx,y,z,. . .

morphisms y xf


f

g



f

g

yz

f

g

= xz

ff

gg


f

g

= xy

f

g


2 t i
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objectsx,y,z,. . .

morphisms y xf


f

g



f

g

yz

f

g

= xz

ff

gg


f

g

= xy

f

g


Associativity requirements for all types of composition:
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1for -morphisms: given w zh y

g x

f we have

(h g) f =h (g f).

for 2-morphisms under vertical composition: given

xy

g

h

f

i

we have ( ) = ( )

for 2-morphisms under horizontal composition: given

xy

f1

g1

yz

f2

g2

zw

f3

g3

we have ()=().


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g x

f we have

(h g) f =h (g f).


xy

g

h

f

i

we have ( ) = ( )


xy

f1

g1

yz

f2

g2

zw

f3

g3

we have ()=().


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ss a y q s a yp s p s


g x

f we have

(h g) f =h (g f).


xy

g

h

f

i

we have ( ) = ( )


xy

f1

g1

yz

f2

g2

zw

f3

g3

we have ()=().


Identity axioms
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y

for 1-morphisms: 1y f =f =f 1x

y x1x x

f = y xf = y y

f x1y

for vertical composition: 1g = = 1f

xy

g

f

g

1g

= x y

f

g

= xyf

f

g

1f

for horizontal composition: 11y== 11x

xy

f

g

yy

1y

1y

11y

= xy

f

g

= xx

1x

1x

11x

xy

f

g


Identity axioms


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y


y x1x x

f = y xf = y y

f x1y


xy

g

f

g

1g

= x y

f

g

= xy

f

f

g

1f


xy

f

g

yy

1y

1y

11y

= xy

f

g

= xx

1x

1x

11x

xy

f

g


Identity axioms


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y


y x1x x

f = y xf = y y

f x1y


xyg

f

g

1g

= x y

f

g

= xy

f

f

g

1f


xy

f

g

yy

1y

1y

11y

= xy

f

g

= xx

1x

1x

11x

xy

f

g

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Examples
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Examples

1 Cat: objects: categories morphisms: functors 2-morphisms: natural transformations

1 Bim: objects: commutative ringsR,S,T, . . . morphisms: (S, R)-bimodules

composition: T STNS R

SMR := T RTNSS SMR

2-morphisms: bimodule homomorphisms

1 (X):

objects: points of a topological space X

morphisms: paths inX

2-morphisms: homotopies between paths


Examples
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p




SMR := T RTNSS SMR


1 (X):





Examples
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p




SMR := T RTNSS SMR


1 (X):





Examples


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p




SMR := T RTNSS SMR


1 (X):





Examples


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p




SMR := T RTNSS SMR


1 (X):





Examples


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composition: T STNS

RSMR

:= T RTNSS SMR


1 (X):

objects: points of a topological space Xmorphisms: paths inX



A tensor category(C, , I)is category with an associative, unital
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te so catego y (C, , ) s catego y t a assoc at e, u tamultiplication for objects and morphisms

y

x

x y

fg

Iis the unit object of C,I x=x=x I.(ThinkVectk, , k)All tensor categories(C, , I)can be thought of as a 2-category

Cwith

objects: just one object, call it

1-morphisms: an objectxof Cis now thought of as a 1-morphism

x

composition : g

f :=

gf

identity 1-morphism: 1 :=

I

2-morphisms: the 1-morphisms from C vertical composition : ordinary composition of 1-morphisms from C horizontal composition: tensor product of 1-morphisms from C


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g y ( , , ) g y ,multiplication for objects and morphisms

y

x

x y

fg


Cwith



x

composition : g

f :=

gf


I



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g y ( , , ) g y ,multiplication for objects and morphisms

y

x

x y

fg


Cwith



x

composition : g

f :=

gf


I



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g y ( , , ) g ymultiplication for objects and morphisms

y

x x

y

fg


Cwith



x

composition : g

f :=

gf


I



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multiplication for objects and morphisms

y

x x

y

fg


Cwith



x

composition : g

f :=

gf


I



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y

x

x

yfg


Cwith



x

composition : g

f :=

gf


I



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y

x

x yfg


Cwith



x

composition : g

f :=

gf


I



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y

x

x yfg


Cwith



x

composition : g

f :=

gf


I



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y

x

x yfg


Cwith



x

composition : g

f :=

gf


I



String diagrams
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Invert the dimensions of the pictures xy

f

g

objects become regions in the plane x or y

morphisms stay 1-dimension, but in an orthogonal direction

y xf y x

f

f

2-morphisms become xy

f

g

y x

g

f


String diagrams
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f

g



y xf y x

f

f


f

g

y x

g

f


String diagrams
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f

g



y xf y x

f

f


f

g

y x

g

f


g

g
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f

y x

f


f

g

yz

f

g

z y x

g

f

g

f

By convention we do not draw identity morphisms or 2-morphisms:

x = x x

1x

1x

y x

f

f

1f = y x

f

f


g

g
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f

y x

f


f

g

yz

f

g

z y x

g

f

g

f


x = x x

1x

1x

y x

f

f

1f = y x

f

f

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g

g


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f

y x

f


f

g

yz

f

g

z y x

g

f

g

f


x = x x

1x

1x

y x

f

f

1f = y x

f

f


y z3g4 z2

g3 z1g2 x

g1


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Ifis a 2-morphism

y w2f3 w1

f2 xf1

thenbecomes the string diagram:

xy

g4 g3 g2 g1

f3 f2 f1

Now lets apply string diagrams to adjoint functors!


y z3g4 z2

g3 z1g2 x

g1
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Ifis a 2-morphism

y w2f3 w1

f2 xf1

thenbecomes the string diagram:

xy

g4 g3 g2 g1

f3 f2 f1

Now lets apply string diagrams to adjoint functors!


Definition

U
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Functors CU

DF

between categories Cand D are adjoint if there

exists a natural bijectionY,X

: HomC

(FY, X)

=Hom

D(Y, GX)

Definition (2)

An adjunction between two categories Cand D consists of functors

C

U

DF

and a natural transformations

1C F U:

U F 1D:

such that for eachX in CandY in D

1UX =U(X) UX or (1U) (1U) =1U

1FY =FY F(Y) or (1F) (1F) =1F

The second definition makes sense in any 2-category.


Definition

U
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Functors C

DF



: HomC

(FY, X)

=Hom

D(Y, GX)

Definition (2)


C

U

DF


1C F U:

U F 1D:


1UX =U(X) UX or (1U) (1U) =1U

1FY =FY F(Y) or (1F) (1F) =1F



Definition

U
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Functors C

DF



: HomC

(FY, X)

=Hom

D(Y, GX)

Definition (2)


C

U

DF


1C F U:

U F 1D:


1UX =U(X) UX or (1U) (1U) =1U

1FY =FY F(Y) or (1F) (1F) =1F



Definition

U
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Functors C

DF



: HomC

(FY, X) = HomD

(Y, GX)

Definition (2)


C

U

DF


1C F U:

U F 1D:


1UX =U(X) UX or (1U) (1U) =1U

1FY =FY F(Y) or (1F) (1F) =1F



Definition

FU

b i C d dj i if h
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Functors C

DF



: HomC

(FY, X) = HomD

(Y, GX)

Definition (2)


C

U

DF


1C F U:

U F 1D:


1UX =U(X) UX or (1U) (1U) =1U

1FY =FY F(Y) or (1F) (1F) =1F



Definition

F tU

b t t i C d D dj i t if th
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Functors CD

F between categories Cand D are adjoint if there


: HomC

(FY, X) = HomD

(Y, GX)

Definition (2)


C

U

DF and a natural transformations

1C F U:

U F 1D:


1UX =U(X) UX or (1U) (1U) =1U

1FY =FY F(Y) or (1F) (1F) =1F



Definition

F t CU

D b t t i C d D dj i t if th
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Functors CD

F between categories Cand D are adjoint if there


: HomC(FY, X) = HomD(Y, GX)

Definition (2)


C

U

DF and a natural transformations

1C F U:

U F 1D:


1UX =U(X) UX or (1U) (1U) =1U

1FY =FY F(Y) or (1F) (1F) =1F


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Definition

An adjunction in a 2-category consists of

bj t d


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objectsxandy

morphisms y xf and x y

u

2-morphisms 1x f u: andu f 1y:

y

x

uf

:=

x x

y

1A

f u

y

x

fu

:=

y y

x

1y

u f

such that the equalities

x

y

f

f

= x y

f

f

x

y

u

u

= y x

u

u

hold.Aaron Lauda (Columbia University) Diagrammatic Algebra December 8th, 2008 14 / 42

Definition


bj t d
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objectsxandy


u


y

x

uf

:=

x x

y

1A

f u

y

x

fu

:=

y y

x

1y

u f


x

y

f

f

= x y

f

f

x

y

u

u

= y x

u

u


Definition


objects x and y
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objectsxandy


u


y

x

uf

:=

x x

y

1A

f u

y

x

fu

:=

y y

x

1y

u f


x

y

f

f

= x y

f

f

x

y

u

u

= y x

u

u


Definition


objects x and y
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objectsxandy


u


y

x

uf

:=

x x

y

1A

f u

y

x

fu

:=

y y

x

1y

u f


x

y

f

f

= x y

f

f

x

y

u

u

= y x

u

u


To see what these diagrams mean we can always convert back into

globular notation:
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g

x

y

u

u

= y x

u

u

y u

x f

y u

x = y

u

u

x

1u

It says that(1u) (1u) =1u


To see what these diagrams mean we can always convert back into

globular notation:
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g

x

y

u

u

= y x

u

u

y u

x f

y u

x = y

u

u

x

1u

It says that(1u) (1u) =1u


Examples of adjunctions

The free group functor is left adjoint to the forgetful functor from
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Groupto Set

In the categoryAb of abelian groups the functor Ais the leftadjoint ofHom(A, )

Given a continuous mapf: X Ybetween topological spaces,then the inverse image functor f1 : Sh(Y) Sh(X)is left adjoint

to the direct image functorf : Sh(X) Sh(Y).Given a inclusion of commutative ringsR A, then the restrictionfunctor

Res: A mod R mod

has a left and right adjoint: the induction functor Ind(M) =A RMand the coinduction functorCoInd(M) =HomR(A, M).

See the Wikipedia page on adjunction for more examples.



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Groupto Set




Res: A mod R mod





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Groupto Set




Res: A mod R mod





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Groupto Set




Res: A mod R mod





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Groupto Set




Res: A mod R mod




Vector spaces and their dualsWhat is an adjunction in Vectk?

V
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A pair of vector spaces:

V := :=

V :=

V

:=

linear mapsV V k, andk V V such that

V

=

V

,

V

=

V

An adjunction in the 2-category Vectkis just a vector space and its

dual!



V
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V := :=

V :=

V

:=


V

=

V

,

V

=

V


dual!



V
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V := :=

V :=

V

:=


V

=

V

,

V

=

V


dual!



V
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V := :=

V :=

V

:=


V

=

V

,

V

=

V


dual!



V
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V := :=

V :=

V

:=


V

=

V

,

V

=

V


dual!


Definition

Biadjoint morphismsin a 2-category consists of

objects x and y
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objectsxandy

morphisms y x

f and x y

u

2-morphismsy

x

uf

y

xfu

x

y

fu

x

yuf

such that

x

y

f

f

= x y

f

f

x

y

u

u

= y x

u

u

y

x

u

u

= y x

u

u

y

x

f

f

= x y

f

f


Definition


objects x and y
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objectsxandy

morphisms y x

f and x y

u

2-morphismsy

x

uf

y

xfu

x

y

fu

x

yuf

such that

x

y

f

f

= x y

f

f

x

y

u

u

= y x

u

u

y

x

u

u

= y x

u

u

y

x

f

f

= x y

f

f


Definition


objects x and y
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objectsxandy

morphisms y x

f and x y

u

2-morphismsy

x

uf

y

xfu

x

y

fu

x

yuf

such that

x

y

f

f

= x y

f

f

x

y

u

u

= y x

u

u

y

x

u

u

= y x

u

u

y

x

f

f

= x y

f

f


Definition


objectsxandy
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j y

morphisms y x

f

and x y

u

2-morphismsy

x

uf

y

xfu

x

y

fu

x

yuf

such that

x

y

f

f

= x y

f

f

x

y

u

u

= y x

u

u

y

x

u

u

= y x

u

u

y

x

f

f

= x y

f

f

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There is a close connection betweenbiadjoints in a 2-category

Frobenius algebras

2 dimensional surfaces


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2-dimensional surfaces

Define the category2PCobof planar 2-dimensional surfaces:objects: disjoint unions of intervals on R

morphisms: 2-dimensional surfaces with boundary intervals:

Definition

A planar 2-dimensional topological quantum field theory is a functor

2PCob Vectk

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Frobenius algebras



87/184




Definition


2PCob Vectk

Aaron Lauda (Columbia University) Diagrammatic Algebra December 8th 2008 19 / 42


Frobenius algebras

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Definition


2PCob Vectk



Frobenius algebras

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Definition


2PCob Vectk



Frobenius algebras

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d e s o a su aces



Definition


2PCob Vectk



Frobenius algebras

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Definition


2PCob Vectk



Frobenius algebras

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Definition


2PCob Vectk


DefinitionA Frobenius algebra is a vector spaceAequipped with

multiplication and unit maps: m: A A A, : k A,
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:=

A A

A

m

:=

A

comulitiplication and counit maps: : A A A, : A k,

:=

A A

A

,

:=

A

satisfying the following axioms:



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:=

A A

A

m

:=

A


:=

A A

A

,

:=

A




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:=

A A

A

m

:=

A


:=

A A

A

,

:=

A




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:=

A A

A

m

:=

A


:=

A A

A

,

:=

A




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We have already seen that

monoidal category 2-category with one object
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but it is also true the other way


Theorem

Given biadjoints xu

yf

in a 2-category K, the objectu f is a

Frobenius algebra in the monoidal categoryHomK(y, y).



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Theorem

Given biadjoints xu

yf





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Theorem

Given biadjoints xu

yf




Proof.

In string diagrams the identity map on uf is

u f
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Try to define a multiplication map ?

u f u f

u f

The unit for this multiplication is : uf 1y. y

xfu

Why is this a unit for multiplication?u f

=

u f

=

u f


Proof.


u f
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u f u f

u f


xfu


=

u f

=

u f


Proof.


u f
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u f u f

u f

:=

u f u f

u f


xfu


=

u f

=

u f


Proof.


u f
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u f u f

u f

:=

u f u f

u f


xfu


=

u f

=

u f

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Proof.


u f


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Try to define a multiplication map :=

u f u f

u f


xfu


=

u f

=

u f


Proof.


u f
http://find/


112/184


u f u f

u f


xfu


=

u f

=

u f


Proof.


u f
http://find/


113/184


u f u f

u f


xfu


=

u f

=

u f


Proof of associativityu f

=

u f
http://find/


114/184

u f u f u f u f u f u f

xy

fu

1y

11y

yx

fu

1y

1fu

= xy

fu

1y

1fu

yx

fu

1y

11y

xy

fu

1y

yx

fu

1y

Just having an adjunction makesufinto a monoid.



=

u f
http://find/


115/184


xy

fu

1y

11y

yx

fu

1y

1fu

= xy

fu

1y

1fu

yx

fu

1y

11y

xy

fu

1y

yx

fu

1y




=

u f
http://find/


116/184


xy

fu

1y

11y

yx

fu

1y

1fu

= xy

fu

1y

1fu

yx

fu

1y

11y

xy

fu

1y

yx

fu

1y




=

u f
http://find/


117/184


xy

fu

1y

11y

yx

fu

1y

1fu

= xy

fu

1y

1fu

yx

fu

1y

11y

xy

fu

1y

yx

fu

1y




=

u f
http://find/


118/184


xy

fu

1y

11y

yx

fu

1y

1fu

= xy

fu

1y

1fu

yx

fu

1y

11y

xy

fu

1y

yx

fu

1y



Comultiplication and counit

Sinceuandf arebiadjointwe can define

comultiplication

u f u f
http://find/


119/184

u f

counit maps

fu

The counit axiom follows from biadjointness.

y

x

u

u

= y x

u

u

y

x

f

f

= x y

f

f

Coassociativity follows from axioms of a 2-category.




comultiplication

u f u f
http://find/


120/184

u f

counit maps

fu


y

x

u

u

= y x

u

u

y

x

f

f

= x y

f

f


http://find/


121/184



comultiplication

u f u f


122/184

u f

counit maps

fu


y

x

u

u

= y x

u

u

y

x

f

f

= x y

f

f





comultiplication

u f u f
http://find/


123/184

u f

counit maps

fu


y

x

u

u

= y x

u

u

y

x

f

f

= x y

f

f



Frobenius identities

= =
http://find/


124/184

= xx

1x

fu

1fu

xx

fu

1x

1fu

= xx

f

g

=

Frobenius relations follow from the interchange law and identity axioms

in a 2-category.



= =
http://find/


125/184

= xx

1x

fu

1fu

xx

fu

1x

1fu

= xx

f

g

=


in a 2-category.



= =
http://find/


126/184

= xx

1x

fu

1fu

xx

fu

1x

1fu

= xx

f

g

=


in a 2-category.



= =
http://find/


127/184

= xx

1x

fu

1fu

xx

fu

1x

1fu

= xx

f

g

=


in a 2-category.



= =
http://find/


128/184

= xx

1x

fu

1fu

xx

fu

1x

1fu

= xx

f

g

=


in a 2-category.



= =
http://find/


129/184

= xx

1x

fu

1fu

xx

fu

1x

1fu

= xx

f

g

=


in a 2-category.


We can see that the strings diagrams for biadjoints, in particular those

inHomK(y, y)look a lot like 2-dimensional planar surfaces.

Theorem

The morphisms in2-PCobare generated by
http://find/


130/184

subject to the relations

= =

= = = =

= =




Theorem

http://find/


131/184


= =

= = = =

= =




Theorem

http://find/


132/184


= =

= = = =

= =


Defining a 2-dimensional planar TQFT is the same as giving a

Frobenius algebra
http://find/


133/184

Every pair of biadjoint morphisms in a 2-category gives a Frobenius

algebra

One can show that the converse is also true, every Frobenius algebra

gives rise to a pair of biadjoint morphisms in some 2-category.

The diagrammatic calculus of string diagrams for 2-categories

illuminates all of these facts.



Frobenius algebra

E i f bi dj i hi i 2 i F b i
http://find/


134/184


algebra







Frobenius algebra

E i f bi dj i t hi i 2 t i F b i
http://find/


135/184


algebra







Frobenius algebra

E i f bi dj i t hi i 2 t i F b i
http://find/


136/184


algebra






Quantum groupsDefinition

The quantum groupUq(sl2)is the associative algebra (with unit) over (q)with generatorsE,F,K,K1 and relations

KK 1 = 1 = K 1K ,
http://find/


137/184

KK 1 K K,KE=q2EK, KF =q2FK,

EF FE= KK1

qq1

Any finite-dimensional representationVhas a weight decomposition

V(n 2)

V(n)

V(n+2)

E

E

F

F V =

n V(n)

KV(n) =q

n

V(n)

Add orthogonal idempotents 1nfor the projectionontoV(n)




KK 1

= 1 = K 1

K ,
http://find/


138/184

KK 1 K K,KE=q2EK, KF =q2FK,

EF FE= KK1

qq1


V(n 2)

V(n)

V(n+2)

E

E

F

F V =

n V(n)

KV(n) =q

n

V(n)





KK1

=1=K1

K,


139/184

,KE=q2EK, KF =q2FK,

EF FE= KK1

qq1


V(n 2)

V(n)

V(n+2)

E

E

F

F V =

n V(n)

KV(n) =q

n

V(n)



CategorificationIgor Frenkel proposed that U(sl2)could be categorified using

Lusztig canonical basis:

E (a)1n := Ea

[a]!1n F(b)1n := F

b

[b]!1n
http://find/


140/184

E 1n: [a]!1n F 1n: [b]!1n

E(a)F(b)1n, n b a

F(b)E(a)1n, n b a

Structure constants are in [q, q1]

U(sl2)is the Grothendieck ring of some higher structure.

Crane and Frenkel conjectured that categorified quantum groups atroots of unity should define 4-dimensional TQFTs sensitive to smooth

structure.




E(a)1n := Ea

[a]!1n F(b)1n := F

b

[b]!1n
http://find/


141/184

n [a]! n n [b]! n

E(a)F(b)1n, n b a

F(b)E(a)1n, n b a




structure.




E(a)1n := Ea

[a]!1n F(b)1n := F

b

[b]!1n
http://find/


142/184

n [a]! n n [b]! n

E(a)F(b)1n, n b a

F(b)E(a)1n, n b a




structure.




E(a)1n:= Ea

[a]!1n F(b)1n:= F

b

[b]!1n
http://find/


143/184

[a]! [b]!

E(a)F(b)1n, n b a

F(b)E(a)1n, n b a




structure.


BeilinsonLusztigMacPherson

UUis a (q)-algebra without unit

Uq(sl2) U

collection ofh l id
http://find/


144/184

1 orthogonal idempotents1nforn

K1n=qn

1n no moreK

E1n=1n+2E=1n+2E1nF1n=1n+2F =1n+2F1n

EF1n FE1n= [n]1n [n] = qnqn

qq

1

Uhas a basis {EaFb1n} forn ,a, b 0




Uq(sl2) U

collection ofth l id t t
http://find/


145/184


K1n=qn

1n no moreK



qq1





Uq(sl2) U

1collection of

th l id t t
http://find/


146/184


K1n=qn

1n no moreK



qq1



BeilinsonLusztigMacPherson U

Uis a (q)-algebra without unit

Uq(sl2) U

1collection of

orthogonal idempotents
http://find/


147/184


K1n=qn

1n no moreK



qq1


http://find/


148/184

Definition

The 2-categoryUconsists of

objects:n

morphisms


149/184

n+ 2 n

n+ 2 nE1n{s}

and n n+2

n 2 nF1n{s}

for alln, s

, together with their composites. We also allow form

direct sums of these 1-morphisms

EaFb1n{s} EcFd1n{s

}


2-morphisms: k-linear combinations of composites of

nn+2

n+2n

n

n

deg2 deg2 deg-2 deg-2

F E E F n n
http://find/


150/184

n

n

F E

E Fdegn+1 deg1-n degn+1 deg1-n

For example

nn

E3F31n{s}

E3F31n{s}

take degrees s

diagramsthis makes the total degree =0


2-morphisms: k-linear combinations of composites of

nn+2

n+2n

n

n

deg2 deg2 deg-2 deg-2

F E E F n n
http://find/


151/184

n

n

F E

E Fdegn+1 deg1-n degn+1 deg1-n

For example

nn

E3F31n{s}

E3F31n{s}

take degrees s

diagramsthis makes the total degree =0

http://find/


152/184
http://find/


153/184

Topological invariance

n+2

n

=

n n+2

=

n+2

n

n

= n =

n
http://find/


154/184

n

=

n =

n

We can define

n:=

n

=

n

n :=

n

=

n



n+2

n

=

n n+2

=

n+2

n

n

= n =

n
http://find/


155/184

n

=

n =

n

We can define

n:=

n

=

n

n :=

n

=

n



n+2

n

=

n n+2

=

n+2

n

n

= n =

n
http://find/


156/184

n

=

n =

n

We can define

n:=

n

=

n

n :=

n

=

n


Positivity of bubbles

All dotted bubbles of negative degree are zero. That is,

deg

n =2(1 n) +2 deg

n =2(1+n) +2

n n
http://find/


157/184

n

= 0 if


158/184

n

= 0 if


159/184

n

= 0 if


160/184

n

= 0 if


161/184

Infinite Grassmannian equation:

n1

n

+

n1+1

n

t+ +

n1+

n

t +

n1

n

+ +

n1+

n

t +

=1.

Analogous to the defining relations inH(Gr(, )), lim Gr(m, 2m), m 2m:

(1+x1t+x2t2 +. . . )(1+y1t+y2t

2 +y3t3 +. . . ) =1


Fake bubbles

ndeg

n 0


162/184

Infinite Grassmannian equation:

n1

n

+

n1+1

n

t+ +

n1+

n

t +

n1

n

+ +

n1+

n

t +

=1.

Analogous to the defining relations inH(Gr(, )), lim Gr(m, 2m), m 2m:

(1+x1t+x2t2 +. . . )(1+y1t+y2t

2 +y3t3 +. . . ) =1

http://find/


163/184

Examples of fake bubbles

Forn 0

n1+0

n

:=1

n

n


164/184

n1+1

:=

n1+1

n1+2

n

:=

n1+2

n

+

n1+1

n1+1

n

n1+j

n 0

= 1+2=j

n1+1

n1+2

n



Forn 0

n1+0

n

:=1

n

n


165/184

n1+1

:=

n1+1

n1+2

n

:=

n1+2

n

+

n1+1

n1+1

n

n1+j

n 0

= 1+2=j

n1+1

n1+2

n



Forn 0

n1+0

n

:=1

n

n
http://find/


166/184

n1+1

:=

n1+1

n1+2

n

:=

n1+2

n

+

n1+1

n1+1

n

n1+j

n 0

= 1+2=j

n1+1

n1+2

n



Forn 0

n1+0

n

:=1

n

n
http://find/


167/184

n1+1

:=

n1+1

n1+2

n

:=

n1+2

n

+

n1+1

n1+1

n

n1+j

n 0

= 1+2=j

n1+1

n1+2

n


Reduction to bubbles

n

= f1+f2

=n

n

(n1)+f1

f2 n

=

g1+g2=n

n

(n1)+g1

g2

EF d i i
http://find/


168/184

EF decomposition

nn =

n+

f1+f2+f3=n1

n

f3

f1

(n1)+f2

nn =

n + g1+g2+g3

=n1 g3

g1

(n1)+g1

n


Reduction to bubbles

n

= f1+f2=n

n

(n1)+f1

f2 n

=

g1+g2=n

n

(n1)+g1

g2

EF d iti
http://find/


169/184

EF decomposition

nn =

n+

f1+f2+f3=n1

n

f3

f1

(n1)+f2

nn =

n + g1+g2+g3

=n1 g3

g1

(n1)+g1

n


Examples of reduction to bubbles

n

= nf=0

n

n1+f

nf

If n > 0n

0
http://find/


170/184

Ifn>0

= 0

Ifn=0 then0

= 0

1

= 0

since deg

0

1

=2(1 0) 2=0 so that

0

1

:=1



n

= nf=0

n

n1+f

nf

If n > 0n

0
http://find/


171/184

Ifn>0

= 0

Ifn=0 then0

= 0

1

= 0

since deg

0

1

=2(1 0) 2=0 so that

0

1

:=1



n

= nf=0

n

n1+f

nf

If n > 0n

= 0
http://find/


172/184

Ifn>0

= 0

Ifn=0 then0

= 0

1

= 0

since deg

0

1

=2(1 0) 2=0 so that

0

1

:=1



n

= nf=0

n

n1+f

nf

If n > 0n

= 0
http://find/


173/184

Ifn>0

= 0

Ifn=0 then0

= 0

1

= 0

since deg

0

1

=2(1 0) 2=0 so that

0

1

:=1



n

= nf=0

n

n1+f

nf

If n > 0n

= 0
http://find/


174/184

Ifn>0

= 0

Ifn=0 then0

= 0

1

= 0

since deg 0

1

=2(1 0) 2=0 so that

0

1

:=1


Ifn= 1 then

n

= nf=0 n n1+f

nf

1

= 1

2

1

1
http://find/


175/184

deg

1

2

=0 and deg

1

1

=1

1

2

:=1 and1

1

= 1

1

1

=

1

+

1

1

In this formula all bubbles are real


Ifn= 1 then

n

= nf=0 n n1+f

nf

1

= 1

2

1

1
http://find/


176/184

deg

1

2

=0 and deg

1

1

=1

1

2

:=1 and1

1

= 1

1

1

=

1

+

1

1



Ifn= 1 then

n

= nf=0 n n1+f

nf

1

= 1

2

1

1
http://find/


177/184

deg

1

2

=0 and deg

1

1

=1

1

2

:=1 and1

1

= 1

1

1

=

1

+

1

1



Ifn= 1 then

n

= nf=0 n n1+f

nf

1

= 1

2

1

1
http://find/


178/184

deg

1

2

=0 and deg

1

1

=1

1

2

:=1 and1

1

= 1

1

1

=

1

+

1

1


http://find/


179/184

TheoremThis graphical calculus is consistent and categorifies U

U =Grothendieck ring/category of this category

Indecomposable 1-morphisms Lusztig canonical basis element

The 2-category Uacts on cohomology of iterated flag varietiesleading to a categorification of the irreducible N-dimensional rep

of U (sl2)


180/184

ofUq(sl2)

Joint with Mikhail Khovanov

This has an extension to U(sln).

A categorification of U+(g)for anyKac-Moody algebragusing a similar

diagrammatic calculus.


TheoremThis graphical calculus is consistent and categorifies U

U =Grothendieck ring/category of this category

Indecomposable 1-morphisms Lusztig canonical basis element

The 2-category Uacts on cohomology of iterated flag varietiesleading to a categorification of the irreducible N-dimensional rep

of U (sl2)
http://find/

Documents

Ide Bikin PDF