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Diagrammatic Algebra
Aaron Lauda
Columbia University
December 8th, 2008
Available athttp://www.math.columbia.edu/ lauda/talks/diagram.pdf
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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.
Aaron Lauda (Columbia University) Diagrammatic Algebra December 8th, 2008 2 / 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.
Aaron Lauda (Columbia University) Diagrammatic Algebra December 8th, 2008 2 / 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.
Aaron Lauda (Columbia University) Diagrammatic Algebra December 8th, 2008 2 / 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.
Aaron Lauda (Columbia University) Diagrammatic Algebra December 8th, 2008 2 / 42
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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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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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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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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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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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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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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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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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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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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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i
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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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2 i
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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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2 t i
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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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2 t i
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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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Associativity requirements for all types of composition:
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Associativity requirements for all types of composition:
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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Associativity requirements for all types of composition:
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Associativity requirements for all types of composition:
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 ()=().
Aaron Lauda (Columbia University) Diagrammatic Algebra December 8th, 2008 5 / 42
Associativity requirements for all types of composition:
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ss a y q s a yp s p s
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 ()=().
Aaron Lauda (Columbia University) Diagrammatic Algebra December 8th, 2008 5 / 42
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
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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
= xy
f
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
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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
xyg
f
g
1g
= x y
f
g
= xy
f
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
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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
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Examples
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p
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
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Examples
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p
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
Aaron Lauda (Columbia University) Diagrammatic Algebra December 8th, 2008 8 / 42
Examples
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p
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
Aaron Lauda (Columbia University) Diagrammatic Algebra December 8th, 2008 8 / 42
Examples
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p
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
Aaron Lauda (Columbia University) Diagrammatic Algebra December 8th, 2008 8 / 42
Examples
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1 Cat: objects: categories morphisms: functors 2-morphisms: natural transformations
1 Bim: objects: commutative ringsR,S,T, . . . morphisms: (S, R)-bimodules
composition: T STNS
RSMR
:= T RTNSS SMR
2-morphisms: bimodule homomorphisms
1 (X):
objects: points of a topological space Xmorphisms: paths inX
2-morphisms: homotopies between paths
Aaron Lauda (Columbia University) Diagrammatic Algebra December 8th, 2008 8 / 42
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
Aaron Lauda (Columbia University) Diagrammatic Algebra December 8th, 2008 9 / 42
A tensor category(C, , I)is category with an associative, unital
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g y ( , , ) g y ,multiplication 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
Aaron Lauda (Columbia University) Diagrammatic Algebra December 8th, 2008 9 / 42
A tensor category(C, , I)is category with an associative, unital
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g y ( , , ) g y ,multiplication 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
Aaron Lauda (Columbia University) Diagrammatic Algebra December 8th, 2008 9 / 42
A tensor category(C, , I)is category with an associative, unital
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g y ( , , ) g ymultiplication 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
Aaron Lauda (Columbia University) Diagrammatic Algebra December 8th, 2008 9 / 42
A tensor category(C, , I)is category with an associative, unital
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multiplication 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
Aaron Lauda (Columbia University) Diagrammatic Algebra December 8th, 2008 9 / 42
A tensor category(C, , I)is category with an associative, unital
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multiplication for objects and morphisms
y
x
x
yfg
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
Aaron Lauda (Columbia University) Diagrammatic Algebra December 8th, 2008 9 / 42
A tensor category(C, , I)is category with an associative, unital
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multiplication for objects and morphisms
y
x
x yfg
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
Aaron Lauda (Columbia University) Diagrammatic Algebra December 8th, 2008 9 / 42
A tensor category(C, , I)is category with an associative, unital
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multiplication for objects and morphisms
y
x
x yfg
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
Aaron Lauda (Columbia University) Diagrammatic Algebra December 8th, 2008 9 / 42
A tensor category(C, , I)is category with an associative, unital
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multiplication for objects and morphisms
y
x
x yfg
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
Aaron Lauda (Columbia University) Diagrammatic Algebra December 8th, 2008 9 / 42
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
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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
Aaron Lauda (Columbia University) Diagrammatic Algebra December 8th, 2008 10 / 42
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
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g
g
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vertical composition xy
f
y x
f
horizontal composition xy
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
Aaron Lauda (Columbia University) Diagrammatic Algebra December 8th, 2008 11 / 42
g
g
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vertical composition xy
f
y x
f
horizontal composition xy
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
Aaron Lauda (Columbia University) Diagrammatic Algebra December 8th, 2008 11 / 42
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g
g
-
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vertical composition xy
f
y x
f
horizontal composition xy
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
Aaron Lauda (Columbia University) Diagrammatic Algebra December 8th, 2008 11 / 42
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!
Aaron Lauda (Columbia University) Diagrammatic Algebra December 8th, 2008 12 / 42
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!
Aaron Lauda (Columbia University) Diagrammatic Algebra December 8th, 2008 12 / 42
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.
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Definition
U
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Functors C
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.
Aaron Lauda (Columbia University) Diagrammatic Algebra December 8th, 2008 13 / 42
Definition
U
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Functors C
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.
Aaron Lauda (Columbia University) Diagrammatic Algebra December 8th, 2008 13 / 42
Definition
U
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Functors C
DF
between categories Cand D are adjoint if there
exists a natural bijectionY,X
: HomC
(FY, X) = HomD
(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.
Aaron Lauda (Columbia University) Diagrammatic Algebra December 8th, 2008 13 / 42
Definition
FU
b i C d dj i if h
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Functors C
DF
between categories Cand D are adjoint if there
exists a natural bijectionY,X
: HomC
(FY, X) = HomD
(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.
Aaron Lauda (Columbia University) Diagrammatic Algebra December 8th, 2008 13 / 42
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
exists a natural bijectionY,X
: HomC
(FY, X) = HomD
(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.
Aaron Lauda (Columbia University) Diagrammatic Algebra December 8th, 2008 13 / 42
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
exists a natural bijectionY,X
: HomC(FY, X) = HomD(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.
Aaron Lauda (Columbia University) Diagrammatic Algebra December 8th, 2008 13 / 42
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-
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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
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
An adjunction in a 2-category consists of
objects x and y
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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
An adjunction in a 2-category consists of
objects x and y
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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
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
Aaron Lauda (Columbia University) Diagrammatic Algebra December 8th, 2008 15 / 42
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
Aaron Lauda (Columbia University) Diagrammatic Algebra December 8th, 2008 15 / 42
Examples of adjunctions
The free group functor is left adjoint to the forgetful functor from
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The free group functor is left adjoint to the forgetful functor from
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.
Aaron Lauda (Columbia University) Diagrammatic Algebra December 8th, 2008 16 / 42
Examples of adjunctions
The free group functor is left adjoint to the forgetful functor from
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The free group functor is left adjoint to the forgetful functor from
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.
Aaron Lauda (Columbia University) Diagrammatic Algebra December 8th, 2008 16 / 42
Examples of adjunctions
The free group functor is left adjoint to the forgetful functor from
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The free group functor is left adjoint to the forgetful functor from
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.
Aaron Lauda (Columbia University) Diagrammatic Algebra December 8th, 2008 16 / 42
Examples of adjunctions
The free group functor is left adjoint to the forgetful functor from
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The free group functor is left adjoint to the forgetful functor from
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.
Aaron Lauda (Columbia University) Diagrammatic Algebra December 8th, 2008 16 / 42
Examples of adjunctions
The free group functor is left adjoint to the forgetful functor from
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The free group functor is left adjoint to the forgetful functor from
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.
Aaron Lauda (Columbia University) Diagrammatic Algebra December 8th, 2008 16 / 42
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!
Aaron Lauda (Columbia University) Diagrammatic Algebra December 8th, 2008 17 / 42
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!
Aaron Lauda (Columbia University) Diagrammatic Algebra December 8th, 2008 17 / 42
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!
Aaron Lauda (Columbia University) Diagrammatic Algebra December 8th, 2008 17 / 42
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!
Aaron Lauda (Columbia University) Diagrammatic Algebra December 8th, 2008 17 / 42
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!
Aaron Lauda (Columbia University) Diagrammatic Algebra December 8th, 2008 17 / 42
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
Aaron Lauda (Columbia University) Diagrammatic Algebra December 8th, 2008 18 / 42
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
Aaron Lauda (Columbia University) Diagrammatic Algebra December 8th, 2008 18 / 42
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
Aaron Lauda (Columbia University) Diagrammatic Algebra December 8th, 2008 18 / 42
Definition
Biadjoint morphismsin a 2-category consists of
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
Aaron Lauda (Columbia University) Diagrammatic Algebra December 8th, 2008 18 / 42
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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
Aaron Lauda (Columbia University) Diagrammatic Algebra December 8th, 2008 19 / 42
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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
Aaron Lauda (Columbia University) Diagrammatic Algebra December 8th 2008 19 / 42
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
Aaron Lauda (Columbia University) Diagrammatic Algebra December 8th 2008 19 / 42
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
Aaron Lauda (Columbia University) Diagrammatic Algebra December 8th 2008 19 / 42
There is a close connection betweenbiadjoints in a 2-category
Frobenius algebras
2-dimensional surfaces
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d e s o a su aces
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
Aaron Lauda (Columbia University) Diagrammatic Algebra December 8th 2008 19 / 42
There is a close connection betweenbiadjoints in a 2-category
Frobenius algebras
2-dimensional surfaces
http://find/ -
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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
Aaron Lauda (Columbia University) Diagrammatic Algebra December 8th 2008 19 / 42
There is a close connection betweenbiadjoints in a 2-category
Frobenius algebras
2-dimensional surfaces
http://find/ -
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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
Aaron Lauda (Columbia University) Diagrammatic Algebra December 8th 2008 19 / 42
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:
Aaron Lauda (Columbia University) Diagrammatic Algebra December 8th 2008 20 / 42
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:
Aaron Lauda (Columbia University) Diagrammatic Algebra December 8th 2008 20 / 42
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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:
Aaron Lauda (Columbia University) Diagrammatic Algebra December 8th 2008 20 / 42
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:
Aaron Lauda (Columbia University) Diagrammatic Algebra December 8th 2008 20 / 42
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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
monoidal category 2-category with one object
Theorem
Given biadjoints xu
yf
in a 2-category K, the objectu f is a
Frobenius algebra in the monoidal categoryHomK(y, y).
Aaron Lauda (Columbia University) Diagrammatic Algebra December 8th, 2008 22 / 42
We have already seen that
monoidal category 2-category with one object
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but it is also true the other way
monoidal category 2-category with one object
Theorem
Given biadjoints xu
yf
in a 2-category K, the objectu f is a
Frobenius algebra in the monoidal categoryHomK(y, y).
Aaron Lauda (Columbia University) Diagrammatic Algebra December 8th, 2008 22 / 42
We have already seen that
monoidal category 2-category with one object
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but it is also true the other way
monoidal category 2-category with one object
Theorem
Given biadjoints xu
yf
in a 2-category K, the objectu f is a
Frobenius algebra in the monoidal categoryHomK(y, y).
Aaron Lauda (Columbia University) Diagrammatic Algebra December 8th, 2008 22 / 42
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
Aaron Lauda (Columbia University) Diagrammatic Algebra December 8th, 2008 23 / 42
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
Aaron Lauda (Columbia University) Diagrammatic Algebra December 8th, 2008 23 / 42
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
:=
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
Aaron Lauda (Columbia University) Diagrammatic Algebra December 8th, 2008 23 / 42
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
:=
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
Aaron Lauda (Columbia University) Diagrammatic Algebra December 8th, 2008 23 / 42
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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
Aaron Lauda (Columbia University) Diagrammatic Algebra December 8th, 2008 23 / 42
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
Aaron Lauda (Columbia University) Diagrammatic Algebra December 8th, 2008 23 / 42
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
Aaron Lauda (Columbia University) Diagrammatic Algebra December 8th, 2008 23 / 42
Proof of associativityu f
=
u f
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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.
Aaron Lauda (Columbia University) Diagrammatic Algebra December 8th, 2008 24 / 42
Proof of associativityu f
=
u f
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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.
Aaron Lauda (Columbia University) Diagrammatic Algebra December 8th, 2008 24 / 42
Proof of associativityu f
=
u f
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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.
Aaron Lauda (Columbia University) Diagrammatic Algebra December 8th, 2008 24 / 42
Proof of associativityu f
=
u f
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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.
Aaron Lauda (Columbia University) Diagrammatic Algebra December 8th, 2008 24 / 42
Proof of associativityu f
=
u f
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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.
Aaron Lauda (Columbia University) Diagrammatic Algebra December 8th, 2008 24 / 42
Comultiplication and counit
Sinceuandf arebiadjointwe can define
comultiplication
u f u f
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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.
Aaron Lauda (Columbia University) Diagrammatic Algebra December 8th, 2008 25 / 42
Comultiplication and counit
Sinceuandf arebiadjointwe can define
comultiplication
u f u f
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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.
Aaron Lauda (Columbia University) Diagrammatic Algebra December 8th, 2008 25 / 42
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Comultiplication and counit
Sinceuandf arebiadjointwe can define
comultiplication
u f u f
-
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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.
Aaron Lauda (Columbia University) Diagrammatic Algebra December 8th, 2008 25 / 42
Comultiplication and counit
Sinceuandf arebiadjointwe can define
comultiplication
u f u f
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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.
Aaron Lauda (Columbia University) Diagrammatic Algebra December 8th, 2008 25 / 42
Frobenius identities
= =
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= 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.
Aaron Lauda (Columbia University) Diagrammatic Algebra December 8th, 2008 26 / 42
Frobenius identities
= =
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= 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.
Aaron Lauda (Columbia University) Diagrammatic Algebra December 8th, 2008 26 / 42
Frobenius identities
= =
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= 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.
Aaron Lauda (Columbia University) Diagrammatic Algebra December 8th, 2008 26 / 42
Frobenius identities
= =
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= 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.
Aaron Lauda (Columbia University) Diagrammatic Algebra December 8th, 2008 26 / 42
Frobenius identities
= =
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= 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.
Aaron Lauda (Columbia University) Diagrammatic Algebra December 8th, 2008 26 / 42
Frobenius identities
= =
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= 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.
Aaron Lauda (Columbia University) Diagrammatic Algebra December 8th, 2008 26 / 42
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
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subject to the relations
= =
= = = =
= =
Aaron Lauda (Columbia University) Diagrammatic Algebra December 8th, 2008 27 / 42
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
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subject to the relations
= =
= = = =
= =
Aaron Lauda (Columbia University) Diagrammatic Algebra December 8th, 2008 27 / 42
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
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subject to the relations
= =
= = = =
= =
Aaron Lauda (Columbia University) Diagrammatic Algebra December 8th, 2008 27 / 42
Defining a 2-dimensional planar TQFT is the same as giving a
Frobenius algebra
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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.
Aaron Lauda (Columbia University) Diagrammatic Algebra December 8th, 2008 28 / 42
Defining a 2-dimensional planar TQFT is the same as giving a
Frobenius algebra
E i f bi dj i hi i 2 i F b i
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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.
Aaron Lauda (Columbia University) Diagrammatic Algebra December 8th, 2008 28 / 42
Defining a 2-dimensional planar TQFT is the same as giving a
Frobenius algebra
E i f bi dj i t hi i 2 t i F b i
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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.
Aaron Lauda (Columbia University) Diagrammatic Algebra December 8th, 2008 28 / 42
Defining a 2-dimensional planar TQFT is the same as giving a
Frobenius algebra
E i f bi dj i t hi i 2 t i F b i
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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.
Aaron Lauda (Columbia University) Diagrammatic Algebra December 8th, 2008 28 / 42
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 ,
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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)
Aaron Lauda (Columbia University) Diagrammatic Algebra December 8th, 2008 29 / 42
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 1
K ,
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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)
Aaron Lauda (Columbia University) Diagrammatic Algebra December 8th, 2008 29 / 42
Quantum groupsDefinition
The quantum groupUq(sl2)is the associative algebra (with unit) over (q)with generatorsE,F,K,K1 and relations
KK1
=1=K1
K,
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,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)
Aaron Lauda (Columbia University) Diagrammatic Algebra December 8th, 2008 29 / 42
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
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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.
Aaron Lauda (Columbia University) Diagrammatic Algebra December 8th, 2008 30 / 42
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
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n [a]! n n [b]! n
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.
Aaron Lauda (Columbia University) Diagrammatic Algebra December 8th, 2008 30 / 42
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
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n [a]! n n [b]! n
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.
Aaron Lauda (Columbia University) Diagrammatic Algebra December 8th, 2008 30 / 42
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
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[a]! [b]!
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.
Aaron Lauda (Columbia University) Diagrammatic Algebra December 8th, 2008 30 / 42
BeilinsonLusztigMacPherson
UUis a (q)-algebra without unit
Uq(sl2) U
collection ofh l id
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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
Aaron Lauda (Columbia University) Diagrammatic Algebra December 8th, 2008 31 / 42
BeilinsonLusztigMacPherson
UUis a (q)-algebra without unit
Uq(sl2) U
collection ofth l id t t
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1 orthogonal idempotents1nforn
K1n=qn
1n no moreK
E1n=1n+2E=1n+2E1nF1n=1n+2F =1n+2F1n
EF1n FE1n= [n]1n [n] = qnqn
qq1
Uhas a basis {EaFb1n} forn ,a, b 0
Aaron Lauda (Columbia University) Diagrammatic Algebra December 8th, 2008 31 / 42
BeilinsonLusztigMacPherson
UUis a (q)-algebra without unit
Uq(sl2) U
1collection of
th l id t t
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1 orthogonal idempotents1nforn
K1n=qn
1n no moreK
E1n=1n+2E=1n+2E1nF1n=1n+2F =1n+2F1n
EF1n FE1n= [n]1n [n] = qnqn
qq1
Uhas a basis {EaFb1n} forn ,a, b 0
Aaron Lauda (Columbia University) Diagrammatic Algebra December 8th, 2008 31 / 42
BeilinsonLusztigMacPherson U
Uis a (q)-algebra without unit
Uq(sl2) U
1collection of
orthogonal idempotents
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1 orthogonal idempotents1nforn
K1n=qn
1n no moreK
E1n=1n+2E=1n+2E1nF1n=1n+2F =1n+2F1n
EF1n FE1n= [n]1n [n] = qnqn
qq1
Uhas a basis {EaFb1n} forn ,a, b 0
Aaron Lauda (Columbia University) Diagrammatic Algebra December 8th, 2008 31 / 42
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Definition
The 2-categoryUconsists of
objects:n
morphisms
-
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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
}
Aaron Lauda (Columbia University) Diagrammatic Algebra December 8th, 2008 32 / 42
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
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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
Aaron Lauda (Columbia University) Diagrammatic Algebra December 8th, 2008 33 / 42
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
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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
Aaron Lauda (Columbia University) Diagrammatic Algebra December 8th, 2008 33 / 42
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Topological invariance
n+2
n
=
n n+2
=
n+2
n
n
= n =
n
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n
=
n =
n
We can define
n:=
n
=
n
n :=
n
=
n
Aaron Lauda (Columbia University) Diagrammatic Algebra December 8th, 2008 35 / 42
Topological invariance
n+2
n
=
n n+2
=
n+2
n
n
= n =
n
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n
=
n =
n
We can define
n:=
n
=
n
n :=
n
=
n
Aaron Lauda (Columbia University) Diagrammatic Algebra December 8th, 2008 35 / 42
Topological invariance
n+2
n
=
n n+2
=
n+2
n
n
= n =
n
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n
=
n =
n
We can define
n:=
n
=
n
n :=
n
=
n
Aaron Lauda (Columbia University) Diagrammatic Algebra December 8th, 2008 35 / 42
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
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n
= 0 if
-
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n
= 0 if
-
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n
= 0 if
-
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n
= 0 if
-
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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
Aaron Lauda (Columbia University) Diagrammatic Algebra December 8th, 2008 37 / 42
Fake bubbles
ndeg
n 0
-
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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
Aaron Lauda (Columbia University) Diagrammatic Algebra December 8th, 2008 37 / 42
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Examples of fake bubbles
Forn 0
n1+0
n
:=1
n
n
-
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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
Aaron Lauda (Columbia University) Diagrammatic Algebra December 8th, 2008 38 / 42
Examples of fake bubbles
Forn 0
n1+0
n
:=1
n
n
http://goforward/http://find/http://goback/ -
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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
Aaron Lauda (Columbia University) Diagrammatic Algebra December 8th, 2008 38 / 42
Examples of fake bubbles
Forn 0
n1+0
n
:=1
n
n
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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
Aaron Lauda (Columbia University) Diagrammatic Algebra December 8th, 2008 38 / 42
Examples of fake bubbles
Forn 0
n1+0
n
:=1
n
n
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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
Aaron Lauda (Columbia University) Diagrammatic Algebra December 8th, 2008 38 / 42
Reduction to bubbles
n
= f1+f2
=n
n
(n1)+f1
f2 n
=
g1+g2=n
n
(n1)+g1
g2
EF d i i
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EF decomposition
nn =
n+
f1+f2+f3=n1
n
f3
f1
(n1)+f2
nn =
n + g1+g2+g3
=n1 g3
g1
(n1)+g1
n
Aaron Lauda (Columbia University) Diagrammatic Algebra December 8th, 2008 39 / 42
Reduction to bubbles
n
= f1+f2=n
n
(n1)+f1
f2 n
=
g1+g2=n
n
(n1)+g1
g2
EF d iti
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EF decomposition
nn =
n+
f1+f2+f3=n1
n
f3
f1
(n1)+f2
nn =
n + g1+g2+g3
=n1 g3
g1
(n1)+g1
n
Aaron Lauda (Columbia University) Diagrammatic Algebra December 8th, 2008 39 / 42
Examples of reduction to bubbles
n
= nf=0
n
n1+f
nf
If n > 0n
0
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Ifn>0
= 0
Ifn=0 then0
= 0
1
= 0
since deg
0
1
=2(1 0) 2=0 so that
0
1
:=1
Aaron Lauda (Columbia University) Diagrammatic Algebra December 8th, 2008 40 / 42
Examples of reduction to bubbles
n
= nf=0
n
n1+f
nf
If n > 0n
0
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Ifn>0
= 0
Ifn=0 then0
= 0
1
= 0
since deg
0
1
=2(1 0) 2=0 so that
0
1
:=1
Aaron Lauda (Columbia University) Diagrammatic Algebra December 8th, 2008 40 / 42
Examples of reduction to bubbles
n
= nf=0
n
n1+f
nf
If n > 0n
= 0
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Ifn>0
= 0
Ifn=0 then0
= 0
1
= 0
since deg
0
1
=2(1 0) 2=0 so that
0
1
:=1
Aaron Lauda (Columbia University) Diagrammatic Algebra December 8th, 2008 40 / 42
Examples of reduction to bubbles
n
= nf=0
n
n1+f
nf
If n > 0n
= 0
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Ifn>0
= 0
Ifn=0 then0
= 0
1
= 0
since deg
0
1
=2(1 0) 2=0 so that
0
1
:=1
Aaron Lauda (Columbia University) Diagrammatic Algebra December 8th, 2008 40 / 42
Examples of reduction to bubbles
n
= nf=0
n
n1+f
nf
If n > 0n
= 0
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Ifn>0
= 0
Ifn=0 then0
= 0
1
= 0
since deg 0
1
=2(1 0) 2=0 so that
0
1
:=1
Aaron Lauda (Columbia University) Diagrammatic Algebra December 8th, 2008 40 / 42
Ifn= 1 then
n
= nf=0 n n1+f
nf
1
= 1
2
1
1
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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
Aaron Lauda (Columbia University) Diagrammatic Algebra December 8th, 2008 41 / 42
Ifn= 1 then
n
= nf=0 n n1+f
nf
1
= 1
2
1
1
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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
Aaron Lauda (Columbia University) Diagrammatic Algebra December 8th, 2008 41 / 42
Ifn= 1 then
n
= nf=0 n n1+f
nf
1
= 1
2
1
1
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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
Aaron Lauda (Columbia University) Diagrammatic Algebra December 8th, 2008 41 / 42
Ifn= 1 then
n
= nf=0 n n1+f
nf
1
= 1
2
1
1
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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
Aaron Lauda (Columbia University) Diagrammatic Algebra December 8th, 2008 41 / 42
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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)
-
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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.
Aaron Lauda (Columbia University) Diagrammatic Algebra December 8th, 2008 42 / 42
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/