Physics Topology Logic and Computationa Rosetta Stone

John Baez and Mike Stay

April 9 2010California State University Fresno

Categories Physics Topology Logic Computationobject system manifold proposition data type

morphism process cobordism proof program

zzzzzzz||

` `

` `

XampY ` Z λxλy x(y)

The Big Idea

Once upon a time mathematics was all about sets

In 1945 Eilenberg and Mac Lane introducedcategories

These put processes on an equal footing with things

In physics we often use categories where

bull objects represent physical systems

bullmorphisms represent physical processes

In classical physics we often use the category Set where

bull an object is a set

bull a morphism is a function

In quantum physics we often use Hilb where

bull an object is a Hilbert space

bull a morphism is a linear operator

A category C consists of

bullA collection of objects If X is an object of C wewrite X isin C

bull For any X Y isin C a set of morphisms f X rarr Y

We require that

bull Every X isin C has an identity morphism1X X rarr X

bullGiven f X rarr Y and g Y rarr Z there is acomposite morphism g f X rarr Z

bull The unit laws hold if f X rarr Y thenf1X = f = 1Y f

bullComposition is associative (hg) f = h(g f )

Feynman used diagrams to describe processes inquantum physics

-------------

HHHHHHHHH

VVVVVVV

LLLLLL

ee

Now we know that these are pictures of morphisms mdashso we can use these diagrams in other contexts

We can draw a morphism

f X rarr Y

like this

f

X

Y

We draw the composite of f X rarr Y and g Y rarr Zlike this

f

g

X

Y

Z

Then the associative law is implicit

f

g

h

X

Y

Z

W

If we draw the identity morphism 1X X rarr X likethis

X

the unit laws are implicit too

For theories with at least 1 dimension of space weneed monoidal categories

Here any pair of morphisms f X rarr Y f prime Xprime rarr Yprimehas a tensor product

f otimes f prime X otimes Xprime rarr Y otimes Yprime

We use this to describe parallel processes

fX

Yf prime

Xprime

Yprime= f otimes f prime

X otimes Xprime

Y otimes YprimeExamplesbull The category Hilb with its usual tensor productotimes

bull The category Set with the cartesian product times

More generally we can draw any morphism

f X1 otimes middot middot middot otimes Xn rarr Y1 otimes middot middot middot otimes Ym

like this

f

X1 X2 X3

Y1 Y2

In physics we use this to depict an interactionbetween particles

By composing and tensoring we can build up biggerdiagrams

f

gh

j

X1 X2 X3 X4

Y1 Y2 Y3 Y4

Z

The monoidal category axioms let us deform thepicture without changing the morphism

f

gh

j

X1 X2 X3 X4

Y1 Y2 Y3 Y4

Z

In theories with least 2 dimensions of space we usebraided monoidal categories We can draw thebraiding

BXY X otimes Y rarr Y otimes X

like this

X Y

It has an inverse drawn like this

XY

Then we have

X

X

Y

Y

= X Y =

Y

Y

X

X

In theories with at least 3 dimensions of space we usesymmetric monoidal categories where

X Y=

YX

The most familiar braided monoidal categories aresymmetric

bull In Set with its cartesian product the standardbraiding is

BXY X times Y rarr Y times X(x y) 7rarr (y x)

bull In Hilb with its usual tensor product the standardbraiding is

BXY X otimes Y rarr Y otimes Xx otimes y 7rarr y otimes x

However in thin films there can be lsquoanyonsrsquo Theseare particle-like excitations described by braided monoidalcategories that are not symmetric

bull Superconducting films the quantum Hall effect

bullGraphene (single-layer graphite) fractional-chargeanyons are possible not yet seen

But therersquos a lot more to this story

THE ROSETTA STONE

Categories Physics Topology Logic Computationobject system manifold proposition data type

morphism process cobordism proof program

zzzzzzz||

` `

` `

XampY ` Z λxλy x(y)

In topology there is a category nCob where

bull objects are (n minus 1)-dimensional manifolds

bullmorphisms are cobordisms

A cobordism f X rarr Y is an n-dimensional manifoldwhose boundary is the disjoint union of X and Y Forexample when n = 2

X

Y

f

We compose cobordisms by gluing the lsquooutputrsquo of oneto the lsquoinputrsquo of the other

X

Y

f

Y

Z

g

X

Z

g f

nCob is a monoidal category We tensor cobordismsby taking their disjoint union

X

Y

f

Xprime

Yprimeg

X otimes Xprime

Y otimes Yprimefotimesg

In fact nCob is a symmetric monoidal categoryX otimes Y

Y otimes X

BXY

In general relativity objects in nCob describe choicesof space while morphisms describe choices of space-time I believe that

Quantum theory will eventually make more sense aspart of a theory of quantum gravity mdash but this can onlybe understood using categories

Why The weird features of quantum theory comefrom the ways that Hilb is less like Set than nCobBut nCob is what we use to describe space and space-time in general relativity

lsquoWeirdrsquo properties of quantum theory correspond tounsurprising properties of spacetime

object morphismbull bull rarr bull

SET set function betweenTHEORY sets

QUANTUM Hilbert space operator betweenTHEORY Hilbert spaces

(state) (process)

GENERAL manifold cobordism betweenRELATIVITY manifolds

(space) (spacetime)

For example Set is lsquocartesianrsquo while nCob and Hilbare not

If a symmetric monoidal category is cartesian youcan do various things including duplication

∆X X rarr X otimes X

In Set we can duplicate as follows

∆X X rarr X times Xx 7rarr (x x)

In Hilb we cannot duplicate the function

X rarr X otimes Xx 7rarr x otimes x

is not linear Itrsquos not a morphism in HilbSo we lsquocannot clone a quantum statersquo

Similarly in nCob there is no duplication despitethis misleading picture for n = 2

X

X otimes X

When n = 1 therersquos typically no cobordism from amanifold X to X otimes X and similarly for n = 4

What about logic and computer science These toostudy categories of things and processes

In proof theory we use categories where

bull an object is a proposition

bull a morphism is a proof

In computer science we use categories where

bull an object is a data type

bull a morphism is a program

In proof theory X ` Y means assuming X we canprove Y But we can also let it mean the set of proofsleading from assumption X to conclusion Y

Since proofs are morphisms we can compose themX ` Y Y ` Z

X ` Z

The identity morphism

X ` X

Logic uses monoidal categories where the tensorproduct is lsquoandrsquo We can tensor propositions andtensor proofs

W ` X Y ` ZWampY ` XampZ

In fact logic uses symmetric monoidal categoriesX ` YampZX ` ZampY

Classical logic is cartesian so it permits duplicationX ` Y

X ` YampY

Linear logic does not

A program that takes data of type X as input andreturns data of type Y can be seen as a morphismf X rarr Y

Categories of data types and programs are monoidalGiven data types X and Xprime there is a data type XotimesXprimeAnd given programs f X rarr Y f Xprime rarr Yprime we canwrite a program f otimes f prime that does these two jobs inparallel

fX

Yf prime

Xprime

Yprime= f otimes f prime

X otimes Xprime

Y otimes Yprime

These categories are are typically symmetric monoidal

X Y

Theyrsquore also cartesian For example we can writeprograms that duplicate data

∆X X rarr X otimes X

But for quantum computation we need programminglanguages that apply to noncartesian categories mdashbecause you canrsquot duplicate quantum data

And in quantum computation using anyons therelevant categories are braided

For more detail read our paper in Bob Coeckersquos forth-coming book New Structures in Physics You can findit now You can find it now on the arXiv

Categories Physics Topology Logic Computationobject system manifold proposition data type

morphism process cobordism proof program

zzzzzzz||

` `

` `

XampY ` Z λxλy x(y)

The Big Idea

Once upon a time mathematics was all about sets

In 1945 Eilenberg and Mac Lane introducedcategories

These put processes on an equal footing with things

In physics we often use categories where

bull objects represent physical systems

bullmorphisms represent physical processes

In classical physics we often use the category Set where

bull an object is a set

bull a morphism is a function

In quantum physics we often use Hilb where

bull an object is a Hilbert space

bull a morphism is a linear operator

A category C consists of

bullA collection of objects If X is an object of C wewrite X isin C

bull For any X Y isin C a set of morphisms f X rarr Y

We require that

bull Every X isin C has an identity morphism1X X rarr X

bullGiven f X rarr Y and g Y rarr Z there is acomposite morphism g f X rarr Z

bull The unit laws hold if f X rarr Y thenf1X = f = 1Y f

bullComposition is associative (hg) f = h(g f )

Feynman used diagrams to describe processes inquantum physics

-------------

HHHHHHHHH

VVVVVVV

LLLLLL

ee

Now we know that these are pictures of morphisms mdashso we can use these diagrams in other contexts

We can draw a morphism

f X rarr Y

like this

f

X

Y

We draw the composite of f X rarr Y and g Y rarr Zlike this

f

g

X

Y

Z

Then the associative law is implicit

f

g

h

X

Y

Z

W

If we draw the identity morphism 1X X rarr X likethis

X

the unit laws are implicit too

For theories with at least 1 dimension of space weneed monoidal categories

Here any pair of morphisms f X rarr Y f prime Xprime rarr Yprimehas a tensor product

f otimes f prime X otimes Xprime rarr Y otimes Yprime

We use this to describe parallel processes

fX

Yf prime

Xprime

Yprime= f otimes f prime

X otimes Xprime

Y otimes YprimeExamplesbull The category Hilb with its usual tensor productotimes

bull The category Set with the cartesian product times

More generally we can draw any morphism

f X1 otimes middot middot middot otimes Xn rarr Y1 otimes middot middot middot otimes Ym

like this

f

X1 X2 X3

Y1 Y2

In physics we use this to depict an interactionbetween particles

By composing and tensoring we can build up biggerdiagrams

f

gh

j

X1 X2 X3 X4

Y1 Y2 Y3 Y4

Z

The monoidal category axioms let us deform thepicture without changing the morphism

f

gh

j

X1 X2 X3 X4

Y1 Y2 Y3 Y4

Z

In theories with least 2 dimensions of space we usebraided monoidal categories We can draw thebraiding

BXY X otimes Y rarr Y otimes X

like this

X Y

It has an inverse drawn like this

XY

Then we have

X

X

Y

Y

= X Y =

Y

Y

X

X

In theories with at least 3 dimensions of space we usesymmetric monoidal categories where

X Y=

YX

The most familiar braided monoidal categories aresymmetric

bull In Set with its cartesian product the standardbraiding is

BXY X times Y rarr Y times X(x y) 7rarr (y x)

bull In Hilb with its usual tensor product the standardbraiding is

BXY X otimes Y rarr Y otimes Xx otimes y 7rarr y otimes x

However in thin films there can be lsquoanyonsrsquo Theseare particle-like excitations described by braided monoidalcategories that are not symmetric

bull Superconducting films the quantum Hall effect

bullGraphene (single-layer graphite) fractional-chargeanyons are possible not yet seen

But therersquos a lot more to this story

THE ROSETTA STONE

Categories Physics Topology Logic Computationobject system manifold proposition data type

morphism process cobordism proof program

zzzzzzz||

` `

` `

XampY ` Z λxλy x(y)

In topology there is a category nCob where

bull objects are (n minus 1)-dimensional manifolds

bullmorphisms are cobordisms

A cobordism f X rarr Y is an n-dimensional manifoldwhose boundary is the disjoint union of X and Y Forexample when n = 2

X

Y

f

We compose cobordisms by gluing the lsquooutputrsquo of oneto the lsquoinputrsquo of the other

X

Y

f

Y

Z

g

X

Z

g f

nCob is a monoidal category We tensor cobordismsby taking their disjoint union

X

Y

f

Xprime

Yprimeg

X otimes Xprime

Y otimes Yprimefotimesg

In fact nCob is a symmetric monoidal categoryX otimes Y

Y otimes X

BXY

In general relativity objects in nCob describe choicesof space while morphisms describe choices of space-time I believe that

Quantum theory will eventually make more sense aspart of a theory of quantum gravity mdash but this can onlybe understood using categories

Why The weird features of quantum theory comefrom the ways that Hilb is less like Set than nCobBut nCob is what we use to describe space and space-time in general relativity

lsquoWeirdrsquo properties of quantum theory correspond tounsurprising properties of spacetime

object morphismbull bull rarr bull

SET set function betweenTHEORY sets

QUANTUM Hilbert space operator betweenTHEORY Hilbert spaces

(state) (process)

GENERAL manifold cobordism betweenRELATIVITY manifolds

(space) (spacetime)

For example Set is lsquocartesianrsquo while nCob and Hilbare not

If a symmetric monoidal category is cartesian youcan do various things including duplication

∆X X rarr X otimes X

In Set we can duplicate as follows

∆X X rarr X times Xx 7rarr (x x)

In Hilb we cannot duplicate the function

X rarr X otimes Xx 7rarr x otimes x

is not linear Itrsquos not a morphism in HilbSo we lsquocannot clone a quantum statersquo

Similarly in nCob there is no duplication despitethis misleading picture for n = 2

X

X otimes X

When n = 1 therersquos typically no cobordism from amanifold X to X otimes X and similarly for n = 4

What about logic and computer science These toostudy categories of things and processes

In proof theory we use categories where

bull an object is a proposition

bull a morphism is a proof

In computer science we use categories where

bull an object is a data type

bull a morphism is a program

In proof theory X ` Y means assuming X we canprove Y But we can also let it mean the set of proofsleading from assumption X to conclusion Y

Since proofs are morphisms we can compose themX ` Y Y ` Z

X ` Z

The identity morphism

X ` X

Logic uses monoidal categories where the tensorproduct is lsquoandrsquo We can tensor propositions andtensor proofs

W ` X Y ` ZWampY ` XampZ

In fact logic uses symmetric monoidal categoriesX ` YampZX ` ZampY

Classical logic is cartesian so it permits duplicationX ` Y

X ` YampY

Linear logic does not

A program that takes data of type X as input andreturns data of type Y can be seen as a morphismf X rarr Y

Categories of data types and programs are monoidalGiven data types X and Xprime there is a data type XotimesXprimeAnd given programs f X rarr Y f Xprime rarr Yprime we canwrite a program f otimes f prime that does these two jobs inparallel

fX

Yf prime

Xprime

Yprime= f otimes f prime

X otimes Xprime

Y otimes Yprime

These categories are are typically symmetric monoidal

X Y

Theyrsquore also cartesian For example we can writeprograms that duplicate data

∆X X rarr X otimes X

But for quantum computation we need programminglanguages that apply to noncartesian categories mdashbecause you canrsquot duplicate quantum data

And in quantum computation using anyons therelevant categories are braided

For more detail read our paper in Bob Coeckersquos forth-coming book New Structures in Physics You can findit now You can find it now on the arXiv

Categories Physics Topology Logic Computationobject system manifold proposition data type

morphism process cobordism proof program

zzzzzzz||

` `

` `

XampY ` Z λxλy x(y)

In physics we often use categories where

bull objects represent physical systems

bullmorphisms represent physical processes

In classical physics we often use the category Set where

bull an object is a set

bull a morphism is a function

In quantum physics we often use Hilb where

bull an object is a Hilbert space

bull a morphism is a linear operator

A category C consists of

bullA collection of objects If X is an object of C wewrite X isin C

bull For any X Y isin C a set of morphisms f X rarr Y

We require that

bull Every X isin C has an identity morphism1X X rarr X

bullGiven f X rarr Y and g Y rarr Z there is acomposite morphism g f X rarr Z

bull The unit laws hold if f X rarr Y thenf1X = f = 1Y f

bullComposition is associative (hg) f = h(g f )

Feynman used diagrams to describe processes inquantum physics

-------------

HHHHHHHHH

VVVVVVV

LLLLLL

ee

Now we know that these are pictures of morphisms mdashso we can use these diagrams in other contexts

We can draw a morphism

f X rarr Y

like this

f

X

Y

We draw the composite of f X rarr Y and g Y rarr Zlike this

f

g

X

Y

Z

Then the associative law is implicit

f

g

h

X

Y

Z

W

If we draw the identity morphism 1X X rarr X likethis

X

the unit laws are implicit too

For theories with at least 1 dimension of space weneed monoidal categories

Here any pair of morphisms f X rarr Y f prime Xprime rarr Yprimehas a tensor product

f otimes f prime X otimes Xprime rarr Y otimes Yprime

We use this to describe parallel processes

fX

Yf prime

Xprime

Yprime= f otimes f prime

X otimes Xprime

Y otimes YprimeExamplesbull The category Hilb with its usual tensor productotimes

bull The category Set with the cartesian product times

More generally we can draw any morphism

f X1 otimes middot middot middot otimes Xn rarr Y1 otimes middot middot middot otimes Ym

like this

f

X1 X2 X3

Y1 Y2

In physics we use this to depict an interactionbetween particles

By composing and tensoring we can build up biggerdiagrams

f

gh

j

X1 X2 X3 X4

Y1 Y2 Y3 Y4

Z

The monoidal category axioms let us deform thepicture without changing the morphism

f

gh

j

X1 X2 X3 X4

Y1 Y2 Y3 Y4

Z

In theories with least 2 dimensions of space we usebraided monoidal categories We can draw thebraiding

BXY X otimes Y rarr Y otimes X

like this

X Y

It has an inverse drawn like this

XY

Then we have

X

X

Y

Y

= X Y =

Y

Y

X

X

In theories with at least 3 dimensions of space we usesymmetric monoidal categories where

X Y=

YX

The most familiar braided monoidal categories aresymmetric

bull In Set with its cartesian product the standardbraiding is

BXY X times Y rarr Y times X(x y) 7rarr (y x)

bull In Hilb with its usual tensor product the standardbraiding is

BXY X otimes Y rarr Y otimes Xx otimes y 7rarr y otimes x

However in thin films there can be lsquoanyonsrsquo Theseare particle-like excitations described by braided monoidalcategories that are not symmetric

bull Superconducting films the quantum Hall effect

bullGraphene (single-layer graphite) fractional-chargeanyons are possible not yet seen

But therersquos a lot more to this story

THE ROSETTA STONE

Categories Physics Topology Logic Computationobject system manifold proposition data type

morphism process cobordism proof program

zzzzzzz||

` `

` `

XampY ` Z λxλy x(y)

In topology there is a category nCob where

bull objects are (n minus 1)-dimensional manifolds

bullmorphisms are cobordisms

A cobordism f X rarr Y is an n-dimensional manifoldwhose boundary is the disjoint union of X and Y Forexample when n = 2

X

Y

f

We compose cobordisms by gluing the lsquooutputrsquo of oneto the lsquoinputrsquo of the other

X

Y

f

Y

Z

g

X

Z

g f

nCob is a monoidal category We tensor cobordismsby taking their disjoint union

X

Y

f

Xprime

Yprimeg

X otimes Xprime

Y otimes Yprimefotimesg

In fact nCob is a symmetric monoidal categoryX otimes Y

Y otimes X

BXY

In general relativity objects in nCob describe choicesof space while morphisms describe choices of space-time I believe that

Quantum theory will eventually make more sense aspart of a theory of quantum gravity mdash but this can onlybe understood using categories

Why The weird features of quantum theory comefrom the ways that Hilb is less like Set than nCobBut nCob is what we use to describe space and space-time in general relativity

lsquoWeirdrsquo properties of quantum theory correspond tounsurprising properties of spacetime

object morphismbull bull rarr bull

SET set function betweenTHEORY sets

QUANTUM Hilbert space operator betweenTHEORY Hilbert spaces

(state) (process)

GENERAL manifold cobordism betweenRELATIVITY manifolds

(space) (spacetime)

For example Set is lsquocartesianrsquo while nCob and Hilbare not

If a symmetric monoidal category is cartesian youcan do various things including duplication

∆X X rarr X otimes X

In Set we can duplicate as follows

∆X X rarr X times Xx 7rarr (x x)

In Hilb we cannot duplicate the function

X rarr X otimes Xx 7rarr x otimes x

is not linear Itrsquos not a morphism in HilbSo we lsquocannot clone a quantum statersquo

Similarly in nCob there is no duplication despitethis misleading picture for n = 2

X

X otimes X

When n = 1 therersquos typically no cobordism from amanifold X to X otimes X and similarly for n = 4

What about logic and computer science These toostudy categories of things and processes

In proof theory we use categories where

bull an object is a proposition

bull a morphism is a proof

In computer science we use categories where

bull an object is a data type

bull a morphism is a program

In proof theory X ` Y means assuming X we canprove Y But we can also let it mean the set of proofsleading from assumption X to conclusion Y

Since proofs are morphisms we can compose themX ` Y Y ` Z

X ` Z

The identity morphism

X ` X

Logic uses monoidal categories where the tensorproduct is lsquoandrsquo We can tensor propositions andtensor proofs

W ` X Y ` ZWampY ` XampZ

In fact logic uses symmetric monoidal categoriesX ` YampZX ` ZampY

Classical logic is cartesian so it permits duplicationX ` Y

X ` YampY

Linear logic does not

A program that takes data of type X as input andreturns data of type Y can be seen as a morphismf X rarr Y

Categories of data types and programs are monoidalGiven data types X and Xprime there is a data type XotimesXprimeAnd given programs f X rarr Y f Xprime rarr Yprime we canwrite a program f otimes f prime that does these two jobs inparallel

fX

Yf prime

Xprime

Yprime= f otimes f prime

X otimes Xprime

Y otimes Yprime

These categories are are typically symmetric monoidal

X Y

Theyrsquore also cartesian For example we can writeprograms that duplicate data

∆X X rarr X otimes X

But for quantum computation we need programminglanguages that apply to noncartesian categories mdashbecause you canrsquot duplicate quantum data

And in quantum computation using anyons therelevant categories are braided

For more detail read our paper in Bob Coeckersquos forth-coming book New Structures in Physics You can findit now You can find it now on the arXiv

Categories Physics Topology Logic Computationobject system manifold proposition data type

morphism process cobordism proof program

zzzzzzz||

` `

` `

XampY ` Z λxλy x(y)

A category C consists of

bullA collection of objects If X is an object of C wewrite X isin C

bull For any X Y isin C a set of morphisms f X rarr Y

We require that

bull Every X isin C has an identity morphism1X X rarr X

bullGiven f X rarr Y and g Y rarr Z there is acomposite morphism g f X rarr Z

bull The unit laws hold if f X rarr Y thenf1X = f = 1Y f

bullComposition is associative (hg) f = h(g f )

Feynman used diagrams to describe processes inquantum physics

-------------

HHHHHHHHH

VVVVVVV

LLLLLL

ee

Now we know that these are pictures of morphisms mdashso we can use these diagrams in other contexts

We can draw a morphism

f X rarr Y

like this

f

X

Y

We draw the composite of f X rarr Y and g Y rarr Zlike this

f

g

X

Y

Z

Then the associative law is implicit

f

g

h

X

Y

Z

W

If we draw the identity morphism 1X X rarr X likethis

X

the unit laws are implicit too

For theories with at least 1 dimension of space weneed monoidal categories

Here any pair of morphisms f X rarr Y f prime Xprime rarr Yprimehas a tensor product

f otimes f prime X otimes Xprime rarr Y otimes Yprime

We use this to describe parallel processes

fX

Yf prime

Xprime

Yprime= f otimes f prime

X otimes Xprime

Y otimes YprimeExamplesbull The category Hilb with its usual tensor productotimes

bull The category Set with the cartesian product times

More generally we can draw any morphism

f X1 otimes middot middot middot otimes Xn rarr Y1 otimes middot middot middot otimes Ym

like this

f

X1 X2 X3

Y1 Y2

In physics we use this to depict an interactionbetween particles

By composing and tensoring we can build up biggerdiagrams

f

gh

j

X1 X2 X3 X4

Y1 Y2 Y3 Y4

Z

The monoidal category axioms let us deform thepicture without changing the morphism

f

gh

j

X1 X2 X3 X4

Y1 Y2 Y3 Y4

Z

In theories with least 2 dimensions of space we usebraided monoidal categories We can draw thebraiding

BXY X otimes Y rarr Y otimes X

like this

X Y

It has an inverse drawn like this

XY

Then we have

X

X

Y

Y

= X Y =

Y

Y

X

X

In theories with at least 3 dimensions of space we usesymmetric monoidal categories where

X Y=

YX

The most familiar braided monoidal categories aresymmetric

bull In Set with its cartesian product the standardbraiding is

BXY X times Y rarr Y times X(x y) 7rarr (y x)

bull In Hilb with its usual tensor product the standardbraiding is

BXY X otimes Y rarr Y otimes Xx otimes y 7rarr y otimes x

However in thin films there can be lsquoanyonsrsquo Theseare particle-like excitations described by braided monoidalcategories that are not symmetric

bull Superconducting films the quantum Hall effect

bullGraphene (single-layer graphite) fractional-chargeanyons are possible not yet seen

But therersquos a lot more to this story

THE ROSETTA STONE

Categories Physics Topology Logic Computationobject system manifold proposition data type

morphism process cobordism proof program

zzzzzzz||

` `

` `

XampY ` Z λxλy x(y)

In topology there is a category nCob where

bull objects are (n minus 1)-dimensional manifolds

bullmorphisms are cobordisms

A cobordism f X rarr Y is an n-dimensional manifoldwhose boundary is the disjoint union of X and Y Forexample when n = 2

X

Y

f

We compose cobordisms by gluing the lsquooutputrsquo of oneto the lsquoinputrsquo of the other

X

Y

f

Y

Z

g

X

Z

g f

nCob is a monoidal category We tensor cobordismsby taking their disjoint union

X

Y

f

Xprime

Yprimeg

X otimes Xprime

Y otimes Yprimefotimesg

In fact nCob is a symmetric monoidal categoryX otimes Y

Y otimes X

BXY

In general relativity objects in nCob describe choicesof space while morphisms describe choices of space-time I believe that

Quantum theory will eventually make more sense aspart of a theory of quantum gravity mdash but this can onlybe understood using categories

Why The weird features of quantum theory comefrom the ways that Hilb is less like Set than nCobBut nCob is what we use to describe space and space-time in general relativity

lsquoWeirdrsquo properties of quantum theory correspond tounsurprising properties of spacetime

object morphismbull bull rarr bull

SET set function betweenTHEORY sets

QUANTUM Hilbert space operator betweenTHEORY Hilbert spaces

(state) (process)

GENERAL manifold cobordism betweenRELATIVITY manifolds

(space) (spacetime)

For example Set is lsquocartesianrsquo while nCob and Hilbare not

If a symmetric monoidal category is cartesian youcan do various things including duplication

∆X X rarr X otimes X

In Set we can duplicate as follows

∆X X rarr X times Xx 7rarr (x x)

In Hilb we cannot duplicate the function

X rarr X otimes Xx 7rarr x otimes x

is not linear Itrsquos not a morphism in HilbSo we lsquocannot clone a quantum statersquo

Similarly in nCob there is no duplication despitethis misleading picture for n = 2

X

X otimes X

When n = 1 therersquos typically no cobordism from amanifold X to X otimes X and similarly for n = 4

What about logic and computer science These toostudy categories of things and processes

In proof theory we use categories where

bull an object is a proposition

bull a morphism is a proof

In computer science we use categories where

bull an object is a data type

bull a morphism is a program

In proof theory X ` Y means assuming X we canprove Y But we can also let it mean the set of proofsleading from assumption X to conclusion Y

Since proofs are morphisms we can compose themX ` Y Y ` Z

X ` Z

The identity morphism

X ` X

Logic uses monoidal categories where the tensorproduct is lsquoandrsquo We can tensor propositions andtensor proofs

W ` X Y ` ZWampY ` XampZ

In fact logic uses symmetric monoidal categoriesX ` YampZX ` ZampY

Classical logic is cartesian so it permits duplicationX ` Y

X ` YampY

Linear logic does not

A program that takes data of type X as input andreturns data of type Y can be seen as a morphismf X rarr Y

Categories of data types and programs are monoidalGiven data types X and Xprime there is a data type XotimesXprimeAnd given programs f X rarr Y f Xprime rarr Yprime we canwrite a program f otimes f prime that does these two jobs inparallel

fX

Yf prime

Xprime

Yprime= f otimes f prime

X otimes Xprime

Y otimes Yprime

These categories are are typically symmetric monoidal

X Y

Theyrsquore also cartesian For example we can writeprograms that duplicate data

∆X X rarr X otimes X

But for quantum computation we need programminglanguages that apply to noncartesian categories mdashbecause you canrsquot duplicate quantum data

And in quantum computation using anyons therelevant categories are braided

For more detail read our paper in Bob Coeckersquos forth-coming book New Structures in Physics You can findit now You can find it now on the arXiv

Categories Physics Topology Logic Computationobject system manifold proposition data type

morphism process cobordism proof program

zzzzzzz||

` `

` `

XampY ` Z λxλy x(y)

Feynman used diagrams to describe processes inquantum physics

-------------

HHHHHHHHH

VVVVVVV

LLLLLL

ee

Now we know that these are pictures of morphisms mdashso we can use these diagrams in other contexts

We can draw a morphism

f X rarr Y

like this

f

X

Y

We draw the composite of f X rarr Y and g Y rarr Zlike this

f

g

X

Y

Z

Then the associative law is implicit

f

g

h

X

Y

Z

W

If we draw the identity morphism 1X X rarr X likethis

X

the unit laws are implicit too

For theories with at least 1 dimension of space weneed monoidal categories

Here any pair of morphisms f X rarr Y f prime Xprime rarr Yprimehas a tensor product

f otimes f prime X otimes Xprime rarr Y otimes Yprime

We use this to describe parallel processes

fX

Yf prime

Xprime

Yprime= f otimes f prime

X otimes Xprime

Y otimes YprimeExamplesbull The category Hilb with its usual tensor productotimes

bull The category Set with the cartesian product times

More generally we can draw any morphism

f X1 otimes middot middot middot otimes Xn rarr Y1 otimes middot middot middot otimes Ym

like this

f

X1 X2 X3

Y1 Y2

In physics we use this to depict an interactionbetween particles

By composing and tensoring we can build up biggerdiagrams

f

gh

j

X1 X2 X3 X4

Y1 Y2 Y3 Y4

Z

The monoidal category axioms let us deform thepicture without changing the morphism

f

gh

j

X1 X2 X3 X4

Y1 Y2 Y3 Y4

Z

In theories with least 2 dimensions of space we usebraided monoidal categories We can draw thebraiding

BXY X otimes Y rarr Y otimes X

like this

X Y

It has an inverse drawn like this

XY

Then we have

X

X

Y

Y

= X Y =

Y

Y

X

X

In theories with at least 3 dimensions of space we usesymmetric monoidal categories where

X Y=

YX

The most familiar braided monoidal categories aresymmetric

bull In Set with its cartesian product the standardbraiding is

BXY X times Y rarr Y times X(x y) 7rarr (y x)

bull In Hilb with its usual tensor product the standardbraiding is

BXY X otimes Y rarr Y otimes Xx otimes y 7rarr y otimes x

However in thin films there can be lsquoanyonsrsquo Theseare particle-like excitations described by braided monoidalcategories that are not symmetric

bull Superconducting films the quantum Hall effect

bullGraphene (single-layer graphite) fractional-chargeanyons are possible not yet seen

But therersquos a lot more to this story

THE ROSETTA STONE

Categories Physics Topology Logic Computationobject system manifold proposition data type

morphism process cobordism proof program

zzzzzzz||

` `

` `

XampY ` Z λxλy x(y)

In topology there is a category nCob where

bull objects are (n minus 1)-dimensional manifolds

bullmorphisms are cobordisms

A cobordism f X rarr Y is an n-dimensional manifoldwhose boundary is the disjoint union of X and Y Forexample when n = 2

X

Y

f

We compose cobordisms by gluing the lsquooutputrsquo of oneto the lsquoinputrsquo of the other

X

Y

f

Y

Z

g

X

Z

g f

nCob is a monoidal category We tensor cobordismsby taking their disjoint union

X

Y

f

Xprime

Yprimeg

X otimes Xprime

Y otimes Yprimefotimesg

In fact nCob is a symmetric monoidal categoryX otimes Y

Y otimes X

BXY

In general relativity objects in nCob describe choicesof space while morphisms describe choices of space-time I believe that

Quantum theory will eventually make more sense aspart of a theory of quantum gravity mdash but this can onlybe understood using categories

Why The weird features of quantum theory comefrom the ways that Hilb is less like Set than nCobBut nCob is what we use to describe space and space-time in general relativity

lsquoWeirdrsquo properties of quantum theory correspond tounsurprising properties of spacetime

object morphismbull bull rarr bull

SET set function betweenTHEORY sets

QUANTUM Hilbert space operator betweenTHEORY Hilbert spaces

(state) (process)

GENERAL manifold cobordism betweenRELATIVITY manifolds

(space) (spacetime)

For example Set is lsquocartesianrsquo while nCob and Hilbare not

If a symmetric monoidal category is cartesian youcan do various things including duplication

∆X X rarr X otimes X

In Set we can duplicate as follows

∆X X rarr X times Xx 7rarr (x x)

In Hilb we cannot duplicate the function

X rarr X otimes Xx 7rarr x otimes x

is not linear Itrsquos not a morphism in HilbSo we lsquocannot clone a quantum statersquo

Similarly in nCob there is no duplication despitethis misleading picture for n = 2

X

X otimes X

When n = 1 therersquos typically no cobordism from amanifold X to X otimes X and similarly for n = 4

What about logic and computer science These toostudy categories of things and processes

In proof theory we use categories where

bull an object is a proposition

bull a morphism is a proof

In computer science we use categories where

bull an object is a data type

bull a morphism is a program

In proof theory X ` Y means assuming X we canprove Y But we can also let it mean the set of proofsleading from assumption X to conclusion Y

Since proofs are morphisms we can compose themX ` Y Y ` Z

X ` Z

The identity morphism

X ` X

Logic uses monoidal categories where the tensorproduct is lsquoandrsquo We can tensor propositions andtensor proofs

W ` X Y ` ZWampY ` XampZ

In fact logic uses symmetric monoidal categoriesX ` YampZX ` ZampY

Classical logic is cartesian so it permits duplicationX ` Y

X ` YampY

Linear logic does not

A program that takes data of type X as input andreturns data of type Y can be seen as a morphismf X rarr Y

Categories of data types and programs are monoidalGiven data types X and Xprime there is a data type XotimesXprimeAnd given programs f X rarr Y f Xprime rarr Yprime we canwrite a program f otimes f prime that does these two jobs inparallel

fX

Yf prime

Xprime

Yprime= f otimes f prime

X otimes Xprime

Y otimes Yprime

These categories are are typically symmetric monoidal

X Y

Theyrsquore also cartesian For example we can writeprograms that duplicate data

∆X X rarr X otimes X

But for quantum computation we need programminglanguages that apply to noncartesian categories mdashbecause you canrsquot duplicate quantum data

And in quantum computation using anyons therelevant categories are braided

For more detail read our paper in Bob Coeckersquos forth-coming book New Structures in Physics You can findit now You can find it now on the arXiv

Categories Physics Topology Logic Computationobject system manifold proposition data type

morphism process cobordism proof program

zzzzzzz||

` `

` `

XampY ` Z λxλy x(y)

We can draw a morphism

f X rarr Y

like this

f

X

Y

We draw the composite of f X rarr Y and g Y rarr Zlike this

f

g

X

Y

Z

Then the associative law is implicit

f

g

h

X

Y

Z

W

If we draw the identity morphism 1X X rarr X likethis

X

the unit laws are implicit too

For theories with at least 1 dimension of space weneed monoidal categories

Here any pair of morphisms f X rarr Y f prime Xprime rarr Yprimehas a tensor product

f otimes f prime X otimes Xprime rarr Y otimes Yprime

We use this to describe parallel processes

fX

Yf prime

Xprime

Yprime= f otimes f prime

X otimes Xprime

Y otimes YprimeExamplesbull The category Hilb with its usual tensor productotimes

bull The category Set with the cartesian product times

More generally we can draw any morphism

f X1 otimes middot middot middot otimes Xn rarr Y1 otimes middot middot middot otimes Ym

like this

f

X1 X2 X3

Y1 Y2

In physics we use this to depict an interactionbetween particles

By composing and tensoring we can build up biggerdiagrams

f

gh

j

X1 X2 X3 X4

Y1 Y2 Y3 Y4

Z

The monoidal category axioms let us deform thepicture without changing the morphism

f

gh

j

X1 X2 X3 X4

Y1 Y2 Y3 Y4

Z

In theories with least 2 dimensions of space we usebraided monoidal categories We can draw thebraiding

BXY X otimes Y rarr Y otimes X

like this

X Y

It has an inverse drawn like this

XY

Then we have

X

X

Y

Y

= X Y =

Y

Y

X

X

In theories with at least 3 dimensions of space we usesymmetric monoidal categories where

X Y=

YX

The most familiar braided monoidal categories aresymmetric

bull In Set with its cartesian product the standardbraiding is

BXY X times Y rarr Y times X(x y) 7rarr (y x)

bull In Hilb with its usual tensor product the standardbraiding is

BXY X otimes Y rarr Y otimes Xx otimes y 7rarr y otimes x

However in thin films there can be lsquoanyonsrsquo Theseare particle-like excitations described by braided monoidalcategories that are not symmetric

bull Superconducting films the quantum Hall effect

bullGraphene (single-layer graphite) fractional-chargeanyons are possible not yet seen

But therersquos a lot more to this story

THE ROSETTA STONE

Categories Physics Topology Logic Computationobject system manifold proposition data type

morphism process cobordism proof program

zzzzzzz||

` `

` `

XampY ` Z λxλy x(y)

In topology there is a category nCob where

bull objects are (n minus 1)-dimensional manifolds

bullmorphisms are cobordisms

A cobordism f X rarr Y is an n-dimensional manifoldwhose boundary is the disjoint union of X and Y Forexample when n = 2

X

Y

f

We compose cobordisms by gluing the lsquooutputrsquo of oneto the lsquoinputrsquo of the other

X

Y

f

Y

Z

g

X

Z

g f

nCob is a monoidal category We tensor cobordismsby taking their disjoint union

X

Y

f

Xprime

Yprimeg

X otimes Xprime

Y otimes Yprimefotimesg

In fact nCob is a symmetric monoidal categoryX otimes Y

Y otimes X

BXY

In general relativity objects in nCob describe choicesof space while morphisms describe choices of space-time I believe that

Quantum theory will eventually make more sense aspart of a theory of quantum gravity mdash but this can onlybe understood using categories

Why The weird features of quantum theory comefrom the ways that Hilb is less like Set than nCobBut nCob is what we use to describe space and space-time in general relativity

lsquoWeirdrsquo properties of quantum theory correspond tounsurprising properties of spacetime

object morphismbull bull rarr bull

SET set function betweenTHEORY sets

QUANTUM Hilbert space operator betweenTHEORY Hilbert spaces

(state) (process)

GENERAL manifold cobordism betweenRELATIVITY manifolds

(space) (spacetime)

For example Set is lsquocartesianrsquo while nCob and Hilbare not

If a symmetric monoidal category is cartesian youcan do various things including duplication

∆X X rarr X otimes X

In Set we can duplicate as follows

∆X X rarr X times Xx 7rarr (x x)

In Hilb we cannot duplicate the function

X rarr X otimes Xx 7rarr x otimes x

is not linear Itrsquos not a morphism in HilbSo we lsquocannot clone a quantum statersquo

Similarly in nCob there is no duplication despitethis misleading picture for n = 2

X

X otimes X

When n = 1 therersquos typically no cobordism from amanifold X to X otimes X and similarly for n = 4

What about logic and computer science These toostudy categories of things and processes

In proof theory we use categories where

bull an object is a proposition

bull a morphism is a proof

In computer science we use categories where

bull an object is a data type

bull a morphism is a program

In proof theory X ` Y means assuming X we canprove Y But we can also let it mean the set of proofsleading from assumption X to conclusion Y

Since proofs are morphisms we can compose themX ` Y Y ` Z

X ` Z

The identity morphism

X ` X

Logic uses monoidal categories where the tensorproduct is lsquoandrsquo We can tensor propositions andtensor proofs

W ` X Y ` ZWampY ` XampZ

In fact logic uses symmetric monoidal categoriesX ` YampZX ` ZampY

Classical logic is cartesian so it permits duplicationX ` Y

X ` YampY

Linear logic does not

A program that takes data of type X as input andreturns data of type Y can be seen as a morphismf X rarr Y

Categories of data types and programs are monoidalGiven data types X and Xprime there is a data type XotimesXprimeAnd given programs f X rarr Y f Xprime rarr Yprime we canwrite a program f otimes f prime that does these two jobs inparallel

fX

Yf prime

Xprime

Yprime= f otimes f prime

X otimes Xprime

Y otimes Yprime

These categories are are typically symmetric monoidal

X Y

Theyrsquore also cartesian For example we can writeprograms that duplicate data

∆X X rarr X otimes X

But for quantum computation we need programminglanguages that apply to noncartesian categories mdashbecause you canrsquot duplicate quantum data

And in quantum computation using anyons therelevant categories are braided

For more detail read our paper in Bob Coeckersquos forth-coming book New Structures in Physics You can findit now You can find it now on the arXiv

Categories Physics Topology Logic Computationobject system manifold proposition data type

morphism process cobordism proof program

zzzzzzz||

` `

` `

XampY ` Z λxλy x(y)

We draw the composite of f X rarr Y and g Y rarr Zlike this

f

g

X

Y

Z

Then the associative law is implicit

f

g

h

X

Y

Z

W

If we draw the identity morphism 1X X rarr X likethis

X

the unit laws are implicit too

For theories with at least 1 dimension of space weneed monoidal categories

Here any pair of morphisms f X rarr Y f prime Xprime rarr Yprimehas a tensor product

f otimes f prime X otimes Xprime rarr Y otimes Yprime

We use this to describe parallel processes

fX

Yf prime

Xprime

Yprime= f otimes f prime

X otimes Xprime

Y otimes YprimeExamplesbull The category Hilb with its usual tensor productotimes

bull The category Set with the cartesian product times

More generally we can draw any morphism

f X1 otimes middot middot middot otimes Xn rarr Y1 otimes middot middot middot otimes Ym

like this

f

X1 X2 X3

Y1 Y2

In physics we use this to depict an interactionbetween particles

By composing and tensoring we can build up biggerdiagrams

f

gh

j

X1 X2 X3 X4

Y1 Y2 Y3 Y4

Z

The monoidal category axioms let us deform thepicture without changing the morphism

f

gh

j

X1 X2 X3 X4

Y1 Y2 Y3 Y4

Z

In theories with least 2 dimensions of space we usebraided monoidal categories We can draw thebraiding

BXY X otimes Y rarr Y otimes X

like this

X Y

It has an inverse drawn like this

XY

Then we have

X

X

Y

Y

= X Y =

Y

Y

X

X

In theories with at least 3 dimensions of space we usesymmetric monoidal categories where

X Y=

YX

The most familiar braided monoidal categories aresymmetric

bull In Set with its cartesian product the standardbraiding is

BXY X times Y rarr Y times X(x y) 7rarr (y x)

bull In Hilb with its usual tensor product the standardbraiding is

BXY X otimes Y rarr Y otimes Xx otimes y 7rarr y otimes x

However in thin films there can be lsquoanyonsrsquo Theseare particle-like excitations described by braided monoidalcategories that are not symmetric

bull Superconducting films the quantum Hall effect

bullGraphene (single-layer graphite) fractional-chargeanyons are possible not yet seen

But therersquos a lot more to this story

THE ROSETTA STONE

Categories Physics Topology Logic Computationobject system manifold proposition data type

morphism process cobordism proof program

zzzzzzz||

` `

` `

XampY ` Z λxλy x(y)

In topology there is a category nCob where

bull objects are (n minus 1)-dimensional manifolds

bullmorphisms are cobordisms

A cobordism f X rarr Y is an n-dimensional manifoldwhose boundary is the disjoint union of X and Y Forexample when n = 2

X

Y

f

We compose cobordisms by gluing the lsquooutputrsquo of oneto the lsquoinputrsquo of the other

X

Y

f

Y

Z

g

X

Z

g f

nCob is a monoidal category We tensor cobordismsby taking their disjoint union

X

Y

f

Xprime

Yprimeg

X otimes Xprime

Y otimes Yprimefotimesg

In fact nCob is a symmetric monoidal categoryX otimes Y

Y otimes X

BXY

In general relativity objects in nCob describe choicesof space while morphisms describe choices of space-time I believe that

Quantum theory will eventually make more sense aspart of a theory of quantum gravity mdash but this can onlybe understood using categories

Why The weird features of quantum theory comefrom the ways that Hilb is less like Set than nCobBut nCob is what we use to describe space and space-time in general relativity

lsquoWeirdrsquo properties of quantum theory correspond tounsurprising properties of spacetime

object morphismbull bull rarr bull

SET set function betweenTHEORY sets

QUANTUM Hilbert space operator betweenTHEORY Hilbert spaces

(state) (process)

GENERAL manifold cobordism betweenRELATIVITY manifolds

(space) (spacetime)

For example Set is lsquocartesianrsquo while nCob and Hilbare not

If a symmetric monoidal category is cartesian youcan do various things including duplication

∆X X rarr X otimes X

In Set we can duplicate as follows

∆X X rarr X times Xx 7rarr (x x)

In Hilb we cannot duplicate the function

X rarr X otimes Xx 7rarr x otimes x

is not linear Itrsquos not a morphism in HilbSo we lsquocannot clone a quantum statersquo

Similarly in nCob there is no duplication despitethis misleading picture for n = 2

X

X otimes X

When n = 1 therersquos typically no cobordism from amanifold X to X otimes X and similarly for n = 4

What about logic and computer science These toostudy categories of things and processes

In proof theory we use categories where

bull an object is a proposition

bull a morphism is a proof

In computer science we use categories where

bull an object is a data type

bull a morphism is a program

In proof theory X ` Y means assuming X we canprove Y But we can also let it mean the set of proofsleading from assumption X to conclusion Y

Since proofs are morphisms we can compose themX ` Y Y ` Z

X ` Z

The identity morphism

X ` X

Logic uses monoidal categories where the tensorproduct is lsquoandrsquo We can tensor propositions andtensor proofs

W ` X Y ` ZWampY ` XampZ

In fact logic uses symmetric monoidal categoriesX ` YampZX ` ZampY

Classical logic is cartesian so it permits duplicationX ` Y

X ` YampY

Linear logic does not

A program that takes data of type X as input andreturns data of type Y can be seen as a morphismf X rarr Y

Categories of data types and programs are monoidalGiven data types X and Xprime there is a data type XotimesXprimeAnd given programs f X rarr Y f Xprime rarr Yprime we canwrite a program f otimes f prime that does these two jobs inparallel

fX

Yf prime

Xprime

Yprime= f otimes f prime

X otimes Xprime

Y otimes Yprime

These categories are are typically symmetric monoidal

X Y

Theyrsquore also cartesian For example we can writeprograms that duplicate data

∆X X rarr X otimes X

But for quantum computation we need programminglanguages that apply to noncartesian categories mdashbecause you canrsquot duplicate quantum data

And in quantum computation using anyons therelevant categories are braided

For more detail read our paper in Bob Coeckersquos forth-coming book New Structures in Physics You can findit now You can find it now on the arXiv

Categories Physics Topology Logic Computationobject system manifold proposition data type

morphism process cobordism proof program

zzzzzzz||

` `

` `

XampY ` Z λxλy x(y)

Then the associative law is implicit

f

g

h

X

Y

Z

W

If we draw the identity morphism 1X X rarr X likethis

X

the unit laws are implicit too

For theories with at least 1 dimension of space weneed monoidal categories

Here any pair of morphisms f X rarr Y f prime Xprime rarr Yprimehas a tensor product

f otimes f prime X otimes Xprime rarr Y otimes Yprime

We use this to describe parallel processes

fX

Yf prime

Xprime

Yprime= f otimes f prime

X otimes Xprime

Y otimes YprimeExamplesbull The category Hilb with its usual tensor productotimes

bull The category Set with the cartesian product times

More generally we can draw any morphism

f X1 otimes middot middot middot otimes Xn rarr Y1 otimes middot middot middot otimes Ym

like this

f

X1 X2 X3

Y1 Y2

In physics we use this to depict an interactionbetween particles

By composing and tensoring we can build up biggerdiagrams

f

gh

j

X1 X2 X3 X4

Y1 Y2 Y3 Y4

Z

The monoidal category axioms let us deform thepicture without changing the morphism

f

gh

j

X1 X2 X3 X4

Y1 Y2 Y3 Y4

Z

In theories with least 2 dimensions of space we usebraided monoidal categories We can draw thebraiding

BXY X otimes Y rarr Y otimes X

like this

X Y

It has an inverse drawn like this

XY

Then we have

X

X

Y

Y

= X Y =

Y

Y

X

X

In theories with at least 3 dimensions of space we usesymmetric monoidal categories where

X Y=

YX

The most familiar braided monoidal categories aresymmetric

bull In Set with its cartesian product the standardbraiding is

BXY X times Y rarr Y times X(x y) 7rarr (y x)

bull In Hilb with its usual tensor product the standardbraiding is

BXY X otimes Y rarr Y otimes Xx otimes y 7rarr y otimes x

However in thin films there can be lsquoanyonsrsquo Theseare particle-like excitations described by braided monoidalcategories that are not symmetric

bull Superconducting films the quantum Hall effect

bullGraphene (single-layer graphite) fractional-chargeanyons are possible not yet seen

But therersquos a lot more to this story

THE ROSETTA STONE

Categories Physics Topology Logic Computationobject system manifold proposition data type

morphism process cobordism proof program

zzzzzzz||

` `

` `

XampY ` Z λxλy x(y)

In topology there is a category nCob where

bull objects are (n minus 1)-dimensional manifolds

bullmorphisms are cobordisms

A cobordism f X rarr Y is an n-dimensional manifoldwhose boundary is the disjoint union of X and Y Forexample when n = 2

X

Y

f

We compose cobordisms by gluing the lsquooutputrsquo of oneto the lsquoinputrsquo of the other

X

Y

f

Y

Z

g

X

Z

g f

nCob is a monoidal category We tensor cobordismsby taking their disjoint union

X

Y

f

Xprime

Yprimeg

X otimes Xprime

Y otimes Yprimefotimesg

In fact nCob is a symmetric monoidal categoryX otimes Y

Y otimes X

BXY

In general relativity objects in nCob describe choicesof space while morphisms describe choices of space-time I believe that

Quantum theory will eventually make more sense aspart of a theory of quantum gravity mdash but this can onlybe understood using categories

Why The weird features of quantum theory comefrom the ways that Hilb is less like Set than nCobBut nCob is what we use to describe space and space-time in general relativity

lsquoWeirdrsquo properties of quantum theory correspond tounsurprising properties of spacetime

object morphismbull bull rarr bull

SET set function betweenTHEORY sets

QUANTUM Hilbert space operator betweenTHEORY Hilbert spaces

(state) (process)

GENERAL manifold cobordism betweenRELATIVITY manifolds

(space) (spacetime)

For example Set is lsquocartesianrsquo while nCob and Hilbare not

If a symmetric monoidal category is cartesian youcan do various things including duplication

∆X X rarr X otimes X

In Set we can duplicate as follows

∆X X rarr X times Xx 7rarr (x x)

In Hilb we cannot duplicate the function

X rarr X otimes Xx 7rarr x otimes x

is not linear Itrsquos not a morphism in HilbSo we lsquocannot clone a quantum statersquo

Similarly in nCob there is no duplication despitethis misleading picture for n = 2

X

X otimes X

When n = 1 therersquos typically no cobordism from amanifold X to X otimes X and similarly for n = 4

What about logic and computer science These toostudy categories of things and processes

In proof theory we use categories where

bull an object is a proposition

bull a morphism is a proof

In computer science we use categories where

bull an object is a data type

bull a morphism is a program

In proof theory X ` Y means assuming X we canprove Y But we can also let it mean the set of proofsleading from assumption X to conclusion Y

Since proofs are morphisms we can compose themX ` Y Y ` Z

X ` Z

The identity morphism

X ` X

Logic uses monoidal categories where the tensorproduct is lsquoandrsquo We can tensor propositions andtensor proofs

W ` X Y ` ZWampY ` XampZ

In fact logic uses symmetric monoidal categoriesX ` YampZX ` ZampY

Classical logic is cartesian so it permits duplicationX ` Y

X ` YampY

Linear logic does not

A program that takes data of type X as input andreturns data of type Y can be seen as a morphismf X rarr Y

Categories of data types and programs are monoidalGiven data types X and Xprime there is a data type XotimesXprimeAnd given programs f X rarr Y f Xprime rarr Yprime we canwrite a program f otimes f prime that does these two jobs inparallel

fX

Yf prime

Xprime

Yprime= f otimes f prime

X otimes Xprime

Y otimes Yprime

These categories are are typically symmetric monoidal

X Y

Theyrsquore also cartesian For example we can writeprograms that duplicate data

∆X X rarr X otimes X

But for quantum computation we need programminglanguages that apply to noncartesian categories mdashbecause you canrsquot duplicate quantum data

And in quantum computation using anyons therelevant categories are braided

For more detail read our paper in Bob Coeckersquos forth-coming book New Structures in Physics You can findit now You can find it now on the arXiv

Categories Physics Topology Logic Computationobject system manifold proposition data type

morphism process cobordism proof program

zzzzzzz||

` `

` `

XampY ` Z λxλy x(y)

For theories with at least 1 dimension of space weneed monoidal categories

Here any pair of morphisms f X rarr Y f prime Xprime rarr Yprimehas a tensor product

f otimes f prime X otimes Xprime rarr Y otimes Yprime

We use this to describe parallel processes

fX

Yf prime

Xprime

Yprime= f otimes f prime

X otimes Xprime

Y otimes YprimeExamplesbull The category Hilb with its usual tensor productotimes

bull The category Set with the cartesian product times

More generally we can draw any morphism

f X1 otimes middot middot middot otimes Xn rarr Y1 otimes middot middot middot otimes Ym

like this

f

X1 X2 X3

Y1 Y2

In physics we use this to depict an interactionbetween particles

By composing and tensoring we can build up biggerdiagrams

f

gh

j

X1 X2 X3 X4

Y1 Y2 Y3 Y4

Z

The monoidal category axioms let us deform thepicture without changing the morphism

f

gh

j

X1 X2 X3 X4

Y1 Y2 Y3 Y4

Z

In theories with least 2 dimensions of space we usebraided monoidal categories We can draw thebraiding

BXY X otimes Y rarr Y otimes X

like this

X Y

It has an inverse drawn like this

XY

Then we have

X

X

Y

Y

= X Y =

Y

Y

X

X

In theories with at least 3 dimensions of space we usesymmetric monoidal categories where

X Y=

YX

The most familiar braided monoidal categories aresymmetric

bull In Set with its cartesian product the standardbraiding is

BXY X times Y rarr Y times X(x y) 7rarr (y x)

bull In Hilb with its usual tensor product the standardbraiding is

BXY X otimes Y rarr Y otimes Xx otimes y 7rarr y otimes x

However in thin films there can be lsquoanyonsrsquo Theseare particle-like excitations described by braided monoidalcategories that are not symmetric

bull Superconducting films the quantum Hall effect

bullGraphene (single-layer graphite) fractional-chargeanyons are possible not yet seen

But therersquos a lot more to this story

THE ROSETTA STONE

Categories Physics Topology Logic Computationobject system manifold proposition data type

morphism process cobordism proof program

zzzzzzz||

` `

` `

XampY ` Z λxλy x(y)

In topology there is a category nCob where

bull objects are (n minus 1)-dimensional manifolds

bullmorphisms are cobordisms

A cobordism f X rarr Y is an n-dimensional manifoldwhose boundary is the disjoint union of X and Y Forexample when n = 2

X

Y

f

We compose cobordisms by gluing the lsquooutputrsquo of oneto the lsquoinputrsquo of the other

X

Y

f

Y

Z

g

X

Z

g f

nCob is a monoidal category We tensor cobordismsby taking their disjoint union

X

Y

f

Xprime

Yprimeg

X otimes Xprime

Y otimes Yprimefotimesg

In fact nCob is a symmetric monoidal categoryX otimes Y

Y otimes X

BXY

In general relativity objects in nCob describe choicesof space while morphisms describe choices of space-time I believe that

Quantum theory will eventually make more sense aspart of a theory of quantum gravity mdash but this can onlybe understood using categories

Why The weird features of quantum theory comefrom the ways that Hilb is less like Set than nCobBut nCob is what we use to describe space and space-time in general relativity

lsquoWeirdrsquo properties of quantum theory correspond tounsurprising properties of spacetime

object morphismbull bull rarr bull

SET set function betweenTHEORY sets

QUANTUM Hilbert space operator betweenTHEORY Hilbert spaces

(state) (process)

GENERAL manifold cobordism betweenRELATIVITY manifolds

(space) (spacetime)

For example Set is lsquocartesianrsquo while nCob and Hilbare not

If a symmetric monoidal category is cartesian youcan do various things including duplication

∆X X rarr X otimes X

In Set we can duplicate as follows

∆X X rarr X times Xx 7rarr (x x)

In Hilb we cannot duplicate the function

X rarr X otimes Xx 7rarr x otimes x

is not linear Itrsquos not a morphism in HilbSo we lsquocannot clone a quantum statersquo

Similarly in nCob there is no duplication despitethis misleading picture for n = 2

X

X otimes X

When n = 1 therersquos typically no cobordism from amanifold X to X otimes X and similarly for n = 4

What about logic and computer science These toostudy categories of things and processes

In proof theory we use categories where

bull an object is a proposition

bull a morphism is a proof

In computer science we use categories where

bull an object is a data type

bull a morphism is a program

In proof theory X ` Y means assuming X we canprove Y But we can also let it mean the set of proofsleading from assumption X to conclusion Y

Since proofs are morphisms we can compose themX ` Y Y ` Z

X ` Z

The identity morphism

X ` X

Logic uses monoidal categories where the tensorproduct is lsquoandrsquo We can tensor propositions andtensor proofs

W ` X Y ` ZWampY ` XampZ

In fact logic uses symmetric monoidal categoriesX ` YampZX ` ZampY

Classical logic is cartesian so it permits duplicationX ` Y

X ` YampY

Linear logic does not

A program that takes data of type X as input andreturns data of type Y can be seen as a morphismf X rarr Y

Categories of data types and programs are monoidalGiven data types X and Xprime there is a data type XotimesXprimeAnd given programs f X rarr Y f Xprime rarr Yprime we canwrite a program f otimes f prime that does these two jobs inparallel

fX

Yf prime

Xprime

Yprime= f otimes f prime

X otimes Xprime

Y otimes Yprime

These categories are are typically symmetric monoidal

X Y

Theyrsquore also cartesian For example we can writeprograms that duplicate data

∆X X rarr X otimes X

But for quantum computation we need programminglanguages that apply to noncartesian categories mdashbecause you canrsquot duplicate quantum data

And in quantum computation using anyons therelevant categories are braided

For more detail read our paper in Bob Coeckersquos forth-coming book New Structures in Physics You can findit now You can find it now on the arXiv

Categories Physics Topology Logic Computationobject system manifold proposition data type

morphism process cobordism proof program

zzzzzzz||

` `

` `

XampY ` Z λxλy x(y)

More generally we can draw any morphism

f X1 otimes middot middot middot otimes Xn rarr Y1 otimes middot middot middot otimes Ym

like this

f

X1 X2 X3

Y1 Y2

In physics we use this to depict an interactionbetween particles

By composing and tensoring we can build up biggerdiagrams

f

gh

j

X1 X2 X3 X4

Y1 Y2 Y3 Y4

Z

The monoidal category axioms let us deform thepicture without changing the morphism

f

gh

j

X1 X2 X3 X4

Y1 Y2 Y3 Y4

Z

In theories with least 2 dimensions of space we usebraided monoidal categories We can draw thebraiding

BXY X otimes Y rarr Y otimes X

like this

X Y

It has an inverse drawn like this

XY

Then we have

X

X

Y

Y

= X Y =

Y

Y

X

X

In theories with at least 3 dimensions of space we usesymmetric monoidal categories where

X Y=

YX

The most familiar braided monoidal categories aresymmetric

bull In Set with its cartesian product the standardbraiding is

BXY X times Y rarr Y times X(x y) 7rarr (y x)

bull In Hilb with its usual tensor product the standardbraiding is

BXY X otimes Y rarr Y otimes Xx otimes y 7rarr y otimes x

However in thin films there can be lsquoanyonsrsquo Theseare particle-like excitations described by braided monoidalcategories that are not symmetric

bull Superconducting films the quantum Hall effect

bullGraphene (single-layer graphite) fractional-chargeanyons are possible not yet seen

But therersquos a lot more to this story

THE ROSETTA STONE

Categories Physics Topology Logic Computationobject system manifold proposition data type

morphism process cobordism proof program

zzzzzzz||

` `

` `

XampY ` Z λxλy x(y)

In topology there is a category nCob where

bull objects are (n minus 1)-dimensional manifolds

bullmorphisms are cobordisms

A cobordism f X rarr Y is an n-dimensional manifoldwhose boundary is the disjoint union of X and Y Forexample when n = 2

X

Y

f

We compose cobordisms by gluing the lsquooutputrsquo of oneto the lsquoinputrsquo of the other

X

Y

f

Y

Z

g

X

Z

g f

nCob is a monoidal category We tensor cobordismsby taking their disjoint union

X

Y

f

Xprime

Yprimeg

X otimes Xprime

Y otimes Yprimefotimesg

In fact nCob is a symmetric monoidal categoryX otimes Y

Y otimes X

BXY

In general relativity objects in nCob describe choicesof space while morphisms describe choices of space-time I believe that

Quantum theory will eventually make more sense aspart of a theory of quantum gravity mdash but this can onlybe understood using categories

Why The weird features of quantum theory comefrom the ways that Hilb is less like Set than nCobBut nCob is what we use to describe space and space-time in general relativity

lsquoWeirdrsquo properties of quantum theory correspond tounsurprising properties of spacetime

object morphismbull bull rarr bull

SET set function betweenTHEORY sets

QUANTUM Hilbert space operator betweenTHEORY Hilbert spaces

(state) (process)

GENERAL manifold cobordism betweenRELATIVITY manifolds

(space) (spacetime)

For example Set is lsquocartesianrsquo while nCob and Hilbare not

If a symmetric monoidal category is cartesian youcan do various things including duplication

∆X X rarr X otimes X

In Set we can duplicate as follows

∆X X rarr X times Xx 7rarr (x x)

In Hilb we cannot duplicate the function

X rarr X otimes Xx 7rarr x otimes x

is not linear Itrsquos not a morphism in HilbSo we lsquocannot clone a quantum statersquo

Similarly in nCob there is no duplication despitethis misleading picture for n = 2

X

X otimes X

When n = 1 therersquos typically no cobordism from amanifold X to X otimes X and similarly for n = 4

What about logic and computer science These toostudy categories of things and processes

In proof theory we use categories where

bull an object is a proposition

bull a morphism is a proof

In computer science we use categories where

bull an object is a data type

bull a morphism is a program

In proof theory X ` Y means assuming X we canprove Y But we can also let it mean the set of proofsleading from assumption X to conclusion Y

Since proofs are morphisms we can compose themX ` Y Y ` Z

X ` Z

The identity morphism

X ` X

Logic uses monoidal categories where the tensorproduct is lsquoandrsquo We can tensor propositions andtensor proofs

W ` X Y ` ZWampY ` XampZ

In fact logic uses symmetric monoidal categoriesX ` YampZX ` ZampY

Classical logic is cartesian so it permits duplicationX ` Y

X ` YampY

Linear logic does not

A program that takes data of type X as input andreturns data of type Y can be seen as a morphismf X rarr Y

Categories of data types and programs are monoidalGiven data types X and Xprime there is a data type XotimesXprimeAnd given programs f X rarr Y f Xprime rarr Yprime we canwrite a program f otimes f prime that does these two jobs inparallel

fX

Yf prime

Xprime

Yprime= f otimes f prime

X otimes Xprime

Y otimes Yprime

These categories are are typically symmetric monoidal

X Y

Theyrsquore also cartesian For example we can writeprograms that duplicate data

∆X X rarr X otimes X

But for quantum computation we need programminglanguages that apply to noncartesian categories mdashbecause you canrsquot duplicate quantum data

And in quantum computation using anyons therelevant categories are braided

For more detail read our paper in Bob Coeckersquos forth-coming book New Structures in Physics You can findit now You can find it now on the arXiv

Categories Physics Topology Logic Computationobject system manifold proposition data type

morphism process cobordism proof program

zzzzzzz||

` `

` `

XampY ` Z λxλy x(y)

By composing and tensoring we can build up biggerdiagrams

f

gh

j

X1 X2 X3 X4

Y1 Y2 Y3 Y4

Z

The monoidal category axioms let us deform thepicture without changing the morphism

f

gh

j

X1 X2 X3 X4

Y1 Y2 Y3 Y4

Z

In theories with least 2 dimensions of space we usebraided monoidal categories We can draw thebraiding

BXY X otimes Y rarr Y otimes X

like this

X Y

It has an inverse drawn like this

XY

Then we have

X

X

Y

Y

= X Y =

Y

Y

X

X

In theories with at least 3 dimensions of space we usesymmetric monoidal categories where

X Y=

YX

The most familiar braided monoidal categories aresymmetric

bull In Set with its cartesian product the standardbraiding is

BXY X times Y rarr Y times X(x y) 7rarr (y x)

bull In Hilb with its usual tensor product the standardbraiding is

BXY X otimes Y rarr Y otimes Xx otimes y 7rarr y otimes x

However in thin films there can be lsquoanyonsrsquo Theseare particle-like excitations described by braided monoidalcategories that are not symmetric

bull Superconducting films the quantum Hall effect

bullGraphene (single-layer graphite) fractional-chargeanyons are possible not yet seen

But therersquos a lot more to this story

THE ROSETTA STONE

Categories Physics Topology Logic Computationobject system manifold proposition data type

morphism process cobordism proof program

zzzzzzz||

` `

` `

XampY ` Z λxλy x(y)

In topology there is a category nCob where

bull objects are (n minus 1)-dimensional manifolds

bullmorphisms are cobordisms

A cobordism f X rarr Y is an n-dimensional manifoldwhose boundary is the disjoint union of X and Y Forexample when n = 2

X

Y

f

We compose cobordisms by gluing the lsquooutputrsquo of oneto the lsquoinputrsquo of the other

X

Y

f

Y

Z

g

X

Z

g f

nCob is a monoidal category We tensor cobordismsby taking their disjoint union

X

Y

f

Xprime

Yprimeg

X otimes Xprime

Y otimes Yprimefotimesg

In fact nCob is a symmetric monoidal categoryX otimes Y

Y otimes X

BXY

In general relativity objects in nCob describe choicesof space while morphisms describe choices of space-time I believe that

Quantum theory will eventually make more sense aspart of a theory of quantum gravity mdash but this can onlybe understood using categories

Why The weird features of quantum theory comefrom the ways that Hilb is less like Set than nCobBut nCob is what we use to describe space and space-time in general relativity

lsquoWeirdrsquo properties of quantum theory correspond tounsurprising properties of spacetime

object morphismbull bull rarr bull

SET set function betweenTHEORY sets

QUANTUM Hilbert space operator betweenTHEORY Hilbert spaces

(state) (process)

GENERAL manifold cobordism betweenRELATIVITY manifolds

(space) (spacetime)

For example Set is lsquocartesianrsquo while nCob and Hilbare not

If a symmetric monoidal category is cartesian youcan do various things including duplication

∆X X rarr X otimes X

In Set we can duplicate as follows

∆X X rarr X times Xx 7rarr (x x)

In Hilb we cannot duplicate the function

X rarr X otimes Xx 7rarr x otimes x

is not linear Itrsquos not a morphism in HilbSo we lsquocannot clone a quantum statersquo

Similarly in nCob there is no duplication despitethis misleading picture for n = 2

X

X otimes X

When n = 1 therersquos typically no cobordism from amanifold X to X otimes X and similarly for n = 4

What about logic and computer science These toostudy categories of things and processes

In proof theory we use categories where

bull an object is a proposition

bull a morphism is a proof

In computer science we use categories where

bull an object is a data type

bull a morphism is a program

In proof theory X ` Y means assuming X we canprove Y But we can also let it mean the set of proofsleading from assumption X to conclusion Y

Since proofs are morphisms we can compose themX ` Y Y ` Z

X ` Z

The identity morphism

X ` X

Logic uses monoidal categories where the tensorproduct is lsquoandrsquo We can tensor propositions andtensor proofs

W ` X Y ` ZWampY ` XampZ

In fact logic uses symmetric monoidal categoriesX ` YampZX ` ZampY

Classical logic is cartesian so it permits duplicationX ` Y

X ` YampY

Linear logic does not

A program that takes data of type X as input andreturns data of type Y can be seen as a morphismf X rarr Y

Categories of data types and programs are monoidalGiven data types X and Xprime there is a data type XotimesXprimeAnd given programs f X rarr Y f Xprime rarr Yprime we canwrite a program f otimes f prime that does these two jobs inparallel

fX

Yf prime

Xprime

Yprime= f otimes f prime

X otimes Xprime

Y otimes Yprime

These categories are are typically symmetric monoidal

X Y

Theyrsquore also cartesian For example we can writeprograms that duplicate data

∆X X rarr X otimes X

But for quantum computation we need programminglanguages that apply to noncartesian categories mdashbecause you canrsquot duplicate quantum data

And in quantum computation using anyons therelevant categories are braided

For more detail read our paper in Bob Coeckersquos forth-coming book New Structures in Physics You can findit now You can find it now on the arXiv

Categories Physics Topology Logic Computationobject system manifold proposition data type

morphism process cobordism proof program

zzzzzzz||

` `

` `

XampY ` Z λxλy x(y)

In theories with least 2 dimensions of space we usebraided monoidal categories We can draw thebraiding

BXY X otimes Y rarr Y otimes X

like this

X Y

It has an inverse drawn like this

XY

Then we have

X

X

Y

Y

= X Y =

Y

Y

X

X

In theories with at least 3 dimensions of space we usesymmetric monoidal categories where

X Y=

YX

The most familiar braided monoidal categories aresymmetric

bull In Set with its cartesian product the standardbraiding is

BXY X times Y rarr Y times X(x y) 7rarr (y x)

bull In Hilb with its usual tensor product the standardbraiding is

BXY X otimes Y rarr Y otimes Xx otimes y 7rarr y otimes x

However in thin films there can be lsquoanyonsrsquo Theseare particle-like excitations described by braided monoidalcategories that are not symmetric

bull Superconducting films the quantum Hall effect

bullGraphene (single-layer graphite) fractional-chargeanyons are possible not yet seen

But therersquos a lot more to this story

THE ROSETTA STONE

Categories Physics Topology Logic Computationobject system manifold proposition data type

morphism process cobordism proof program

zzzzzzz||

` `

` `

XampY ` Z λxλy x(y)

In topology there is a category nCob where

bull objects are (n minus 1)-dimensional manifolds

bullmorphisms are cobordisms

A cobordism f X rarr Y is an n-dimensional manifoldwhose boundary is the disjoint union of X and Y Forexample when n = 2

X

Y

f

We compose cobordisms by gluing the lsquooutputrsquo of oneto the lsquoinputrsquo of the other

X

Y

f

Y

Z

g

X

Z

g f

nCob is a monoidal category We tensor cobordismsby taking their disjoint union

X

Y

f

Xprime

Yprimeg

X otimes Xprime

Y otimes Yprimefotimesg

In fact nCob is a symmetric monoidal categoryX otimes Y

Y otimes X

BXY

In general relativity objects in nCob describe choicesof space while morphisms describe choices of space-time I believe that

Quantum theory will eventually make more sense aspart of a theory of quantum gravity mdash but this can onlybe understood using categories

Why The weird features of quantum theory comefrom the ways that Hilb is less like Set than nCobBut nCob is what we use to describe space and space-time in general relativity

lsquoWeirdrsquo properties of quantum theory correspond tounsurprising properties of spacetime

object morphismbull bull rarr bull

SET set function betweenTHEORY sets

QUANTUM Hilbert space operator betweenTHEORY Hilbert spaces

(state) (process)

GENERAL manifold cobordism betweenRELATIVITY manifolds

(space) (spacetime)

For example Set is lsquocartesianrsquo while nCob and Hilbare not

If a symmetric monoidal category is cartesian youcan do various things including duplication

∆X X rarr X otimes X

In Set we can duplicate as follows

∆X X rarr X times Xx 7rarr (x x)

In Hilb we cannot duplicate the function

X rarr X otimes Xx 7rarr x otimes x

is not linear Itrsquos not a morphism in HilbSo we lsquocannot clone a quantum statersquo

Similarly in nCob there is no duplication despitethis misleading picture for n = 2

X

X otimes X

When n = 1 therersquos typically no cobordism from amanifold X to X otimes X and similarly for n = 4

What about logic and computer science These toostudy categories of things and processes

In proof theory we use categories where

bull an object is a proposition

bull a morphism is a proof

In computer science we use categories where

bull an object is a data type

bull a morphism is a program

In proof theory X ` Y means assuming X we canprove Y But we can also let it mean the set of proofsleading from assumption X to conclusion Y

Since proofs are morphisms we can compose themX ` Y Y ` Z

X ` Z

The identity morphism

X ` X

Logic uses monoidal categories where the tensorproduct is lsquoandrsquo We can tensor propositions andtensor proofs

W ` X Y ` ZWampY ` XampZ

In fact logic uses symmetric monoidal categoriesX ` YampZX ` ZampY

Classical logic is cartesian so it permits duplicationX ` Y

X ` YampY

Linear logic does not

A program that takes data of type X as input andreturns data of type Y can be seen as a morphismf X rarr Y

Categories of data types and programs are monoidalGiven data types X and Xprime there is a data type XotimesXprimeAnd given programs f X rarr Y f Xprime rarr Yprime we canwrite a program f otimes f prime that does these two jobs inparallel

fX

Yf prime

Xprime

Yprime= f otimes f prime

X otimes Xprime

Y otimes Yprime

These categories are are typically symmetric monoidal

X Y

Theyrsquore also cartesian For example we can writeprograms that duplicate data

∆X X rarr X otimes X

But for quantum computation we need programminglanguages that apply to noncartesian categories mdashbecause you canrsquot duplicate quantum data

And in quantum computation using anyons therelevant categories are braided

For more detail read our paper in Bob Coeckersquos forth-coming book New Structures in Physics You can findit now You can find it now on the arXiv

Categories Physics Topology Logic Computationobject system manifold proposition data type

morphism process cobordism proof program

zzzzzzz||

` `

` `

XampY ` Z λxλy x(y)

Then we have

X

X

Y

Y

= X Y =

Y

Y

X

X

In theories with at least 3 dimensions of space we usesymmetric monoidal categories where

X Y=

YX

The most familiar braided monoidal categories aresymmetric

bull In Set with its cartesian product the standardbraiding is

BXY X times Y rarr Y times X(x y) 7rarr (y x)

bull In Hilb with its usual tensor product the standardbraiding is

BXY X otimes Y rarr Y otimes Xx otimes y 7rarr y otimes x

However in thin films there can be lsquoanyonsrsquo Theseare particle-like excitations described by braided monoidalcategories that are not symmetric

bull Superconducting films the quantum Hall effect

bullGraphene (single-layer graphite) fractional-chargeanyons are possible not yet seen

But therersquos a lot more to this story

THE ROSETTA STONE

Categories Physics Topology Logic Computationobject system manifold proposition data type

morphism process cobordism proof program

zzzzzzz||

` `

` `

XampY ` Z λxλy x(y)

In topology there is a category nCob where

bull objects are (n minus 1)-dimensional manifolds

bullmorphisms are cobordisms

A cobordism f X rarr Y is an n-dimensional manifoldwhose boundary is the disjoint union of X and Y Forexample when n = 2

X

Y

f

We compose cobordisms by gluing the lsquooutputrsquo of oneto the lsquoinputrsquo of the other

X

Y

f

Y

Z

g

X

Z

g f

nCob is a monoidal category We tensor cobordismsby taking their disjoint union

X

Y

f

Xprime

Yprimeg

X otimes Xprime

Y otimes Yprimefotimesg

In fact nCob is a symmetric monoidal categoryX otimes Y

Y otimes X

BXY

In general relativity objects in nCob describe choicesof space while morphisms describe choices of space-time I believe that

Quantum theory will eventually make more sense aspart of a theory of quantum gravity mdash but this can onlybe understood using categories

Why The weird features of quantum theory comefrom the ways that Hilb is less like Set than nCobBut nCob is what we use to describe space and space-time in general relativity

lsquoWeirdrsquo properties of quantum theory correspond tounsurprising properties of spacetime

object morphismbull bull rarr bull

SET set function betweenTHEORY sets

QUANTUM Hilbert space operator betweenTHEORY Hilbert spaces

(state) (process)

GENERAL manifold cobordism betweenRELATIVITY manifolds

(space) (spacetime)

For example Set is lsquocartesianrsquo while nCob and Hilbare not

If a symmetric monoidal category is cartesian youcan do various things including duplication

∆X X rarr X otimes X

In Set we can duplicate as follows

∆X X rarr X times Xx 7rarr (x x)

In Hilb we cannot duplicate the function

X rarr X otimes Xx 7rarr x otimes x

is not linear Itrsquos not a morphism in HilbSo we lsquocannot clone a quantum statersquo

Similarly in nCob there is no duplication despitethis misleading picture for n = 2

X

X otimes X

When n = 1 therersquos typically no cobordism from amanifold X to X otimes X and similarly for n = 4

What about logic and computer science These toostudy categories of things and processes

In proof theory we use categories where

bull an object is a proposition

bull a morphism is a proof

In computer science we use categories where

bull an object is a data type

bull a morphism is a program

In proof theory X ` Y means assuming X we canprove Y But we can also let it mean the set of proofsleading from assumption X to conclusion Y

Since proofs are morphisms we can compose themX ` Y Y ` Z

X ` Z

The identity morphism

X ` X

Logic uses monoidal categories where the tensorproduct is lsquoandrsquo We can tensor propositions andtensor proofs

W ` X Y ` ZWampY ` XampZ

In fact logic uses symmetric monoidal categoriesX ` YampZX ` ZampY

Classical logic is cartesian so it permits duplicationX ` Y

X ` YampY

Linear logic does not

A program that takes data of type X as input andreturns data of type Y can be seen as a morphismf X rarr Y

Categories of data types and programs are monoidalGiven data types X and Xprime there is a data type XotimesXprimeAnd given programs f X rarr Y f Xprime rarr Yprime we canwrite a program f otimes f prime that does these two jobs inparallel

fX

Yf prime

Xprime

Yprime= f otimes f prime

X otimes Xprime

Y otimes Yprime

These categories are are typically symmetric monoidal

X Y

Theyrsquore also cartesian For example we can writeprograms that duplicate data

∆X X rarr X otimes X

But for quantum computation we need programminglanguages that apply to noncartesian categories mdashbecause you canrsquot duplicate quantum data

And in quantum computation using anyons therelevant categories are braided

For more detail read our paper in Bob Coeckersquos forth-coming book New Structures in Physics You can findit now You can find it now on the arXiv

Categories Physics Topology Logic Computationobject system manifold proposition data type

morphism process cobordism proof program

zzzzzzz||

` `

` `

XampY ` Z λxλy x(y)

The most familiar braided monoidal categories aresymmetric

bull In Set with its cartesian product the standardbraiding is

BXY X times Y rarr Y times X(x y) 7rarr (y x)

bull In Hilb with its usual tensor product the standardbraiding is

BXY X otimes Y rarr Y otimes Xx otimes y 7rarr y otimes x

However in thin films there can be lsquoanyonsrsquo Theseare particle-like excitations described by braided monoidalcategories that are not symmetric

bull Superconducting films the quantum Hall effect

bullGraphene (single-layer graphite) fractional-chargeanyons are possible not yet seen

But therersquos a lot more to this story

THE ROSETTA STONE

Categories Physics Topology Logic Computationobject system manifold proposition data type

morphism process cobordism proof program

zzzzzzz||

` `

` `

XampY ` Z λxλy x(y)

In topology there is a category nCob where

bull objects are (n minus 1)-dimensional manifolds

bullmorphisms are cobordisms

A cobordism f X rarr Y is an n-dimensional manifoldwhose boundary is the disjoint union of X and Y Forexample when n = 2

X

Y

f

We compose cobordisms by gluing the lsquooutputrsquo of oneto the lsquoinputrsquo of the other

X

Y

f

Y

Z

g

X

Z

g f

nCob is a monoidal category We tensor cobordismsby taking their disjoint union

X

Y

f

Xprime

Yprimeg

X otimes Xprime

Y otimes Yprimefotimesg

In fact nCob is a symmetric monoidal categoryX otimes Y

Y otimes X

BXY

In general relativity objects in nCob describe choicesof space while morphisms describe choices of space-time I believe that

Quantum theory will eventually make more sense aspart of a theory of quantum gravity mdash but this can onlybe understood using categories

Why The weird features of quantum theory comefrom the ways that Hilb is less like Set than nCobBut nCob is what we use to describe space and space-time in general relativity

lsquoWeirdrsquo properties of quantum theory correspond tounsurprising properties of spacetime

object morphismbull bull rarr bull

SET set function betweenTHEORY sets

QUANTUM Hilbert space operator betweenTHEORY Hilbert spaces

(state) (process)

GENERAL manifold cobordism betweenRELATIVITY manifolds

(space) (spacetime)

For example Set is lsquocartesianrsquo while nCob and Hilbare not

If a symmetric monoidal category is cartesian youcan do various things including duplication

∆X X rarr X otimes X

In Set we can duplicate as follows

∆X X rarr X times Xx 7rarr (x x)

In Hilb we cannot duplicate the function

X rarr X otimes Xx 7rarr x otimes x

is not linear Itrsquos not a morphism in HilbSo we lsquocannot clone a quantum statersquo

Similarly in nCob there is no duplication despitethis misleading picture for n = 2

X

X otimes X

When n = 1 therersquos typically no cobordism from amanifold X to X otimes X and similarly for n = 4

What about logic and computer science These toostudy categories of things and processes

In proof theory we use categories where

bull an object is a proposition

bull a morphism is a proof

In computer science we use categories where

bull an object is a data type

bull a morphism is a program

In proof theory X ` Y means assuming X we canprove Y But we can also let it mean the set of proofsleading from assumption X to conclusion Y

Since proofs are morphisms we can compose themX ` Y Y ` Z

X ` Z

The identity morphism

X ` X

Logic uses monoidal categories where the tensorproduct is lsquoandrsquo We can tensor propositions andtensor proofs

W ` X Y ` ZWampY ` XampZ

In fact logic uses symmetric monoidal categoriesX ` YampZX ` ZampY

Classical logic is cartesian so it permits duplicationX ` Y

X ` YampY

Linear logic does not

A program that takes data of type X as input andreturns data of type Y can be seen as a morphismf X rarr Y

Categories of data types and programs are monoidalGiven data types X and Xprime there is a data type XotimesXprimeAnd given programs f X rarr Y f Xprime rarr Yprime we canwrite a program f otimes f prime that does these two jobs inparallel

fX

Yf prime

Xprime

Yprime= f otimes f prime

X otimes Xprime

Y otimes Yprime

These categories are are typically symmetric monoidal

X Y

Theyrsquore also cartesian For example we can writeprograms that duplicate data

∆X X rarr X otimes X

But for quantum computation we need programminglanguages that apply to noncartesian categories mdashbecause you canrsquot duplicate quantum data

And in quantum computation using anyons therelevant categories are braided

For more detail read our paper in Bob Coeckersquos forth-coming book New Structures in Physics You can findit now You can find it now on the arXiv

Categories Physics Topology Logic Computationobject system manifold proposition data type

morphism process cobordism proof program

zzzzzzz||

` `

` `

XampY ` Z λxλy x(y)

However in thin films there can be lsquoanyonsrsquo Theseare particle-like excitations described by braided monoidalcategories that are not symmetric

bull Superconducting films the quantum Hall effect

bullGraphene (single-layer graphite) fractional-chargeanyons are possible not yet seen

But therersquos a lot more to this story

THE ROSETTA STONE

Categories Physics Topology Logic Computationobject system manifold proposition data type

morphism process cobordism proof program

zzzzzzz||

` `

` `

XampY ` Z λxλy x(y)

In topology there is a category nCob where

bull objects are (n minus 1)-dimensional manifolds

bullmorphisms are cobordisms

A cobordism f X rarr Y is an n-dimensional manifoldwhose boundary is the disjoint union of X and Y Forexample when n = 2

X

Y

f

We compose cobordisms by gluing the lsquooutputrsquo of oneto the lsquoinputrsquo of the other

X

Y

f

Y

Z

g

X

Z

g f

nCob is a monoidal category We tensor cobordismsby taking their disjoint union

X

Y

f

Xprime

Yprimeg

X otimes Xprime

Y otimes Yprimefotimesg

In fact nCob is a symmetric monoidal categoryX otimes Y

Y otimes X

BXY

In general relativity objects in nCob describe choicesof space while morphisms describe choices of space-time I believe that

Quantum theory will eventually make more sense aspart of a theory of quantum gravity mdash but this can onlybe understood using categories

Why The weird features of quantum theory comefrom the ways that Hilb is less like Set than nCobBut nCob is what we use to describe space and space-time in general relativity

lsquoWeirdrsquo properties of quantum theory correspond tounsurprising properties of spacetime

object morphismbull bull rarr bull

SET set function betweenTHEORY sets

QUANTUM Hilbert space operator betweenTHEORY Hilbert spaces

(state) (process)

GENERAL manifold cobordism betweenRELATIVITY manifolds

(space) (spacetime)

For example Set is lsquocartesianrsquo while nCob and Hilbare not

If a symmetric monoidal category is cartesian youcan do various things including duplication

∆X X rarr X otimes X

In Set we can duplicate as follows

∆X X rarr X times Xx 7rarr (x x)

In Hilb we cannot duplicate the function

X rarr X otimes Xx 7rarr x otimes x

is not linear Itrsquos not a morphism in HilbSo we lsquocannot clone a quantum statersquo

Similarly in nCob there is no duplication despitethis misleading picture for n = 2

X

X otimes X

When n = 1 therersquos typically no cobordism from amanifold X to X otimes X and similarly for n = 4

What about logic and computer science These toostudy categories of things and processes

In proof theory we use categories where

bull an object is a proposition

bull a morphism is a proof

In computer science we use categories where

bull an object is a data type

bull a morphism is a program

In proof theory X ` Y means assuming X we canprove Y But we can also let it mean the set of proofsleading from assumption X to conclusion Y

Since proofs are morphisms we can compose themX ` Y Y ` Z

X ` Z

The identity morphism

X ` X

Logic uses monoidal categories where the tensorproduct is lsquoandrsquo We can tensor propositions andtensor proofs

W ` X Y ` ZWampY ` XampZ

In fact logic uses symmetric monoidal categoriesX ` YampZX ` ZampY

Classical logic is cartesian so it permits duplicationX ` Y

X ` YampY

Linear logic does not

A program that takes data of type X as input andreturns data of type Y can be seen as a morphismf X rarr Y

Categories of data types and programs are monoidalGiven data types X and Xprime there is a data type XotimesXprimeAnd given programs f X rarr Y f Xprime rarr Yprime we canwrite a program f otimes f prime that does these two jobs inparallel

fX

Yf prime

Xprime

Yprime= f otimes f prime

X otimes Xprime

Y otimes Yprime

These categories are are typically symmetric monoidal

X Y

Theyrsquore also cartesian For example we can writeprograms that duplicate data

∆X X rarr X otimes X

But for quantum computation we need programminglanguages that apply to noncartesian categories mdashbecause you canrsquot duplicate quantum data

And in quantum computation using anyons therelevant categories are braided

For more detail read our paper in Bob Coeckersquos forth-coming book New Structures in Physics You can findit now You can find it now on the arXiv

Categories Physics Topology Logic Computationobject system manifold proposition data type

morphism process cobordism proof program

zzzzzzz||

` `

` `

XampY ` Z λxλy x(y)

But therersquos a lot more to this story

THE ROSETTA STONE

Categories Physics Topology Logic Computationobject system manifold proposition data type

morphism process cobordism proof program

zzzzzzz||

` `

` `

XampY ` Z λxλy x(y)

In topology there is a category nCob where

bull objects are (n minus 1)-dimensional manifolds

bullmorphisms are cobordisms

A cobordism f X rarr Y is an n-dimensional manifoldwhose boundary is the disjoint union of X and Y Forexample when n = 2

X

Y

f

We compose cobordisms by gluing the lsquooutputrsquo of oneto the lsquoinputrsquo of the other

X

Y

f

Y

Z

g

X

Z

g f

nCob is a monoidal category We tensor cobordismsby taking their disjoint union

X

Y

f

Xprime

Yprimeg

X otimes Xprime

Y otimes Yprimefotimesg

In fact nCob is a symmetric monoidal categoryX otimes Y

Y otimes X

BXY

In general relativity objects in nCob describe choicesof space while morphisms describe choices of space-time I believe that

Quantum theory will eventually make more sense aspart of a theory of quantum gravity mdash but this can onlybe understood using categories

Why The weird features of quantum theory comefrom the ways that Hilb is less like Set than nCobBut nCob is what we use to describe space and space-time in general relativity

lsquoWeirdrsquo properties of quantum theory correspond tounsurprising properties of spacetime

object morphismbull bull rarr bull

SET set function betweenTHEORY sets

QUANTUM Hilbert space operator betweenTHEORY Hilbert spaces

(state) (process)

GENERAL manifold cobordism betweenRELATIVITY manifolds

(space) (spacetime)

For example Set is lsquocartesianrsquo while nCob and Hilbare not

If a symmetric monoidal category is cartesian youcan do various things including duplication

∆X X rarr X otimes X

In Set we can duplicate as follows

∆X X rarr X times Xx 7rarr (x x)

In Hilb we cannot duplicate the function

X rarr X otimes Xx 7rarr x otimes x

is not linear Itrsquos not a morphism in HilbSo we lsquocannot clone a quantum statersquo

Similarly in nCob there is no duplication despitethis misleading picture for n = 2

X

X otimes X

When n = 1 therersquos typically no cobordism from amanifold X to X otimes X and similarly for n = 4

What about logic and computer science These toostudy categories of things and processes

In proof theory we use categories where

bull an object is a proposition

bull a morphism is a proof

In computer science we use categories where

bull an object is a data type

bull a morphism is a program

In proof theory X ` Y means assuming X we canprove Y But we can also let it mean the set of proofsleading from assumption X to conclusion Y

Since proofs are morphisms we can compose themX ` Y Y ` Z

X ` Z

The identity morphism

X ` X

Logic uses monoidal categories where the tensorproduct is lsquoandrsquo We can tensor propositions andtensor proofs

W ` X Y ` ZWampY ` XampZ

In fact logic uses symmetric monoidal categoriesX ` YampZX ` ZampY

Classical logic is cartesian so it permits duplicationX ` Y

X ` YampY

Linear logic does not

A program that takes data of type X as input andreturns data of type Y can be seen as a morphismf X rarr Y

Categories of data types and programs are monoidalGiven data types X and Xprime there is a data type XotimesXprimeAnd given programs f X rarr Y f Xprime rarr Yprime we canwrite a program f otimes f prime that does these two jobs inparallel

fX

Yf prime

Xprime

Yprime= f otimes f prime

X otimes Xprime

Y otimes Yprime

These categories are are typically symmetric monoidal

X Y

Theyrsquore also cartesian For example we can writeprograms that duplicate data

∆X X rarr X otimes X

But for quantum computation we need programminglanguages that apply to noncartesian categories mdashbecause you canrsquot duplicate quantum data

And in quantum computation using anyons therelevant categories are braided

For more detail read our paper in Bob Coeckersquos forth-coming book New Structures in Physics You can findit now You can find it now on the arXiv

Categories Physics Topology Logic Computationobject system manifold proposition data type

morphism process cobordism proof program

zzzzzzz||

` `

` `

XampY ` Z λxλy x(y)

In topology there is a category nCob where

bull objects are (n minus 1)-dimensional manifolds

bullmorphisms are cobordisms

A cobordism f X rarr Y is an n-dimensional manifoldwhose boundary is the disjoint union of X and Y Forexample when n = 2

X

Y

f

We compose cobordisms by gluing the lsquooutputrsquo of oneto the lsquoinputrsquo of the other

X

Y

f

Y

Z

g

X

Z

g f

nCob is a monoidal category We tensor cobordismsby taking their disjoint union

X

Y

f

Xprime

Yprimeg

X otimes Xprime

Y otimes Yprimefotimesg

In fact nCob is a symmetric monoidal categoryX otimes Y

Y otimes X

BXY

In general relativity objects in nCob describe choicesof space while morphisms describe choices of space-time I believe that

Quantum theory will eventually make more sense aspart of a theory of quantum gravity mdash but this can onlybe understood using categories

Why The weird features of quantum theory comefrom the ways that Hilb is less like Set than nCobBut nCob is what we use to describe space and space-time in general relativity

lsquoWeirdrsquo properties of quantum theory correspond tounsurprising properties of spacetime

object morphismbull bull rarr bull

SET set function betweenTHEORY sets

QUANTUM Hilbert space operator betweenTHEORY Hilbert spaces

(state) (process)

GENERAL manifold cobordism betweenRELATIVITY manifolds

(space) (spacetime)

For example Set is lsquocartesianrsquo while nCob and Hilbare not

If a symmetric monoidal category is cartesian youcan do various things including duplication

∆X X rarr X otimes X

In Set we can duplicate as follows

∆X X rarr X times Xx 7rarr (x x)

In Hilb we cannot duplicate the function

X rarr X otimes Xx 7rarr x otimes x

is not linear Itrsquos not a morphism in HilbSo we lsquocannot clone a quantum statersquo

Similarly in nCob there is no duplication despitethis misleading picture for n = 2

X

X otimes X

When n = 1 therersquos typically no cobordism from amanifold X to X otimes X and similarly for n = 4

What about logic and computer science These toostudy categories of things and processes

In proof theory we use categories where

bull an object is a proposition

bull a morphism is a proof

In computer science we use categories where

bull an object is a data type

bull a morphism is a program

In proof theory X ` Y means assuming X we canprove Y But we can also let it mean the set of proofsleading from assumption X to conclusion Y

Since proofs are morphisms we can compose themX ` Y Y ` Z

X ` Z

The identity morphism

X ` X

Logic uses monoidal categories where the tensorproduct is lsquoandrsquo We can tensor propositions andtensor proofs

W ` X Y ` ZWampY ` XampZ

In fact logic uses symmetric monoidal categoriesX ` YampZX ` ZampY

Classical logic is cartesian so it permits duplicationX ` Y

X ` YampY

Linear logic does not

A program that takes data of type X as input andreturns data of type Y can be seen as a morphismf X rarr Y

Categories of data types and programs are monoidalGiven data types X and Xprime there is a data type XotimesXprimeAnd given programs f X rarr Y f Xprime rarr Yprime we canwrite a program f otimes f prime that does these two jobs inparallel

fX

Yf prime

Xprime

Yprime= f otimes f prime

X otimes Xprime

Y otimes Yprime

These categories are are typically symmetric monoidal

X Y

Theyrsquore also cartesian For example we can writeprograms that duplicate data

∆X X rarr X otimes X

But for quantum computation we need programminglanguages that apply to noncartesian categories mdashbecause you canrsquot duplicate quantum data

And in quantum computation using anyons therelevant categories are braided

For more detail read our paper in Bob Coeckersquos forth-coming book New Structures in Physics You can findit now You can find it now on the arXiv

Categories Physics Topology Logic Computationobject system manifold proposition data type

morphism process cobordism proof program

zzzzzzz||

` `

` `

XampY ` Z λxλy x(y)

We compose cobordisms by gluing the lsquooutputrsquo of oneto the lsquoinputrsquo of the other

X

Y

f

Y

Z

g

X

Z

g f

nCob is a monoidal category We tensor cobordismsby taking their disjoint union

X

Y

f

Xprime

Yprimeg

X otimes Xprime

Y otimes Yprimefotimesg

In fact nCob is a symmetric monoidal categoryX otimes Y

Y otimes X

BXY

In general relativity objects in nCob describe choicesof space while morphisms describe choices of space-time I believe that

Quantum theory will eventually make more sense aspart of a theory of quantum gravity mdash but this can onlybe understood using categories

Why The weird features of quantum theory comefrom the ways that Hilb is less like Set than nCobBut nCob is what we use to describe space and space-time in general relativity

lsquoWeirdrsquo properties of quantum theory correspond tounsurprising properties of spacetime

object morphismbull bull rarr bull

SET set function betweenTHEORY sets

QUANTUM Hilbert space operator betweenTHEORY Hilbert spaces

(state) (process)

GENERAL manifold cobordism betweenRELATIVITY manifolds

(space) (spacetime)

For example Set is lsquocartesianrsquo while nCob and Hilbare not

If a symmetric monoidal category is cartesian youcan do various things including duplication

∆X X rarr X otimes X

In Set we can duplicate as follows

∆X X rarr X times Xx 7rarr (x x)

In Hilb we cannot duplicate the function

X rarr X otimes Xx 7rarr x otimes x

is not linear Itrsquos not a morphism in HilbSo we lsquocannot clone a quantum statersquo

Similarly in nCob there is no duplication despitethis misleading picture for n = 2

X

X otimes X

When n = 1 therersquos typically no cobordism from amanifold X to X otimes X and similarly for n = 4

What about logic and computer science These toostudy categories of things and processes

In proof theory we use categories where

bull an object is a proposition

bull a morphism is a proof

In computer science we use categories where

bull an object is a data type

bull a morphism is a program

In proof theory X ` Y means assuming X we canprove Y But we can also let it mean the set of proofsleading from assumption X to conclusion Y

Since proofs are morphisms we can compose themX ` Y Y ` Z

X ` Z

The identity morphism

X ` X

Logic uses monoidal categories where the tensorproduct is lsquoandrsquo We can tensor propositions andtensor proofs

W ` X Y ` ZWampY ` XampZ

In fact logic uses symmetric monoidal categoriesX ` YampZX ` ZampY

Classical logic is cartesian so it permits duplicationX ` Y

X ` YampY

Linear logic does not

A program that takes data of type X as input andreturns data of type Y can be seen as a morphismf X rarr Y

Categories of data types and programs are monoidalGiven data types X and Xprime there is a data type XotimesXprimeAnd given programs f X rarr Y f Xprime rarr Yprime we canwrite a program f otimes f prime that does these two jobs inparallel

fX

Yf prime

Xprime

Yprime= f otimes f prime

X otimes Xprime

Y otimes Yprime

These categories are are typically symmetric monoidal

X Y

Theyrsquore also cartesian For example we can writeprograms that duplicate data

∆X X rarr X otimes X

But for quantum computation we need programminglanguages that apply to noncartesian categories mdashbecause you canrsquot duplicate quantum data

And in quantum computation using anyons therelevant categories are braided

For more detail read our paper in Bob Coeckersquos forth-coming book New Structures in Physics You can findit now You can find it now on the arXiv

Categories Physics Topology Logic Computationobject system manifold proposition data type

morphism process cobordism proof program

zzzzzzz||

` `

` `

XampY ` Z λxλy x(y)

nCob is a monoidal category We tensor cobordismsby taking their disjoint union

X

Y

f

Xprime

Yprimeg

X otimes Xprime

Y otimes Yprimefotimesg

In fact nCob is a symmetric monoidal categoryX otimes Y

Y otimes X

BXY

In general relativity objects in nCob describe choicesof space while morphisms describe choices of space-time I believe that

Quantum theory will eventually make more sense aspart of a theory of quantum gravity mdash but this can onlybe understood using categories

Why The weird features of quantum theory comefrom the ways that Hilb is less like Set than nCobBut nCob is what we use to describe space and space-time in general relativity

lsquoWeirdrsquo properties of quantum theory correspond tounsurprising properties of spacetime

object morphismbull bull rarr bull

SET set function betweenTHEORY sets

QUANTUM Hilbert space operator betweenTHEORY Hilbert spaces

(state) (process)

GENERAL manifold cobordism betweenRELATIVITY manifolds

(space) (spacetime)

For example Set is lsquocartesianrsquo while nCob and Hilbare not

If a symmetric monoidal category is cartesian youcan do various things including duplication

∆X X rarr X otimes X

In Set we can duplicate as follows

∆X X rarr X times Xx 7rarr (x x)

In Hilb we cannot duplicate the function

X rarr X otimes Xx 7rarr x otimes x

is not linear Itrsquos not a morphism in HilbSo we lsquocannot clone a quantum statersquo

Similarly in nCob there is no duplication despitethis misleading picture for n = 2

X

X otimes X

When n = 1 therersquos typically no cobordism from amanifold X to X otimes X and similarly for n = 4

What about logic and computer science These toostudy categories of things and processes

In proof theory we use categories where

bull an object is a proposition

bull a morphism is a proof

In computer science we use categories where

bull an object is a data type

bull a morphism is a program

In proof theory X ` Y means assuming X we canprove Y But we can also let it mean the set of proofsleading from assumption X to conclusion Y

Since proofs are morphisms we can compose themX ` Y Y ` Z

X ` Z

The identity morphism

X ` X

Logic uses monoidal categories where the tensorproduct is lsquoandrsquo We can tensor propositions andtensor proofs

W ` X Y ` ZWampY ` XampZ

In fact logic uses symmetric monoidal categoriesX ` YampZX ` ZampY

Classical logic is cartesian so it permits duplicationX ` Y

X ` YampY

Linear logic does not

A program that takes data of type X as input andreturns data of type Y can be seen as a morphismf X rarr Y

Categories of data types and programs are monoidalGiven data types X and Xprime there is a data type XotimesXprimeAnd given programs f X rarr Y f Xprime rarr Yprime we canwrite a program f otimes f prime that does these two jobs inparallel

fX

Yf prime

Xprime

Yprime= f otimes f prime

X otimes Xprime

Y otimes Yprime

These categories are are typically symmetric monoidal

X Y

Theyrsquore also cartesian For example we can writeprograms that duplicate data

∆X X rarr X otimes X

But for quantum computation we need programminglanguages that apply to noncartesian categories mdashbecause you canrsquot duplicate quantum data

And in quantum computation using anyons therelevant categories are braided

For more detail read our paper in Bob Coeckersquos forth-coming book New Structures in Physics You can findit now You can find it now on the arXiv

Categories Physics Topology Logic Computationobject system manifold proposition data type

morphism process cobordism proof program

zzzzzzz||

` `

` `

XampY ` Z λxλy x(y)

In fact nCob is a symmetric monoidal categoryX otimes Y

Y otimes X

BXY

In general relativity objects in nCob describe choicesof space while morphisms describe choices of space-time I believe that

Quantum theory will eventually make more sense aspart of a theory of quantum gravity mdash but this can onlybe understood using categories

Why The weird features of quantum theory comefrom the ways that Hilb is less like Set than nCobBut nCob is what we use to describe space and space-time in general relativity

lsquoWeirdrsquo properties of quantum theory correspond tounsurprising properties of spacetime

object morphismbull bull rarr bull

SET set function betweenTHEORY sets

QUANTUM Hilbert space operator betweenTHEORY Hilbert spaces

(state) (process)

GENERAL manifold cobordism betweenRELATIVITY manifolds

(space) (spacetime)

For example Set is lsquocartesianrsquo while nCob and Hilbare not

If a symmetric monoidal category is cartesian youcan do various things including duplication

∆X X rarr X otimes X

In Set we can duplicate as follows

∆X X rarr X times Xx 7rarr (x x)

In Hilb we cannot duplicate the function

X rarr X otimes Xx 7rarr x otimes x

is not linear Itrsquos not a morphism in HilbSo we lsquocannot clone a quantum statersquo

Similarly in nCob there is no duplication despitethis misleading picture for n = 2

X

X otimes X

When n = 1 therersquos typically no cobordism from amanifold X to X otimes X and similarly for n = 4

What about logic and computer science These toostudy categories of things and processes

In proof theory we use categories where

bull an object is a proposition

bull a morphism is a proof

In computer science we use categories where

bull an object is a data type

bull a morphism is a program

In proof theory X ` Y means assuming X we canprove Y But we can also let it mean the set of proofsleading from assumption X to conclusion Y

Since proofs are morphisms we can compose themX ` Y Y ` Z

X ` Z

The identity morphism

X ` X

Logic uses monoidal categories where the tensorproduct is lsquoandrsquo We can tensor propositions andtensor proofs

W ` X Y ` ZWampY ` XampZ

In fact logic uses symmetric monoidal categoriesX ` YampZX ` ZampY

Classical logic is cartesian so it permits duplicationX ` Y

X ` YampY

Linear logic does not

A program that takes data of type X as input andreturns data of type Y can be seen as a morphismf X rarr Y

Categories of data types and programs are monoidalGiven data types X and Xprime there is a data type XotimesXprimeAnd given programs f X rarr Y f Xprime rarr Yprime we canwrite a program f otimes f prime that does these two jobs inparallel

fX

Yf prime

Xprime

Yprime= f otimes f prime

X otimes Xprime

Y otimes Yprime

These categories are are typically symmetric monoidal

X Y

Theyrsquore also cartesian For example we can writeprograms that duplicate data

∆X X rarr X otimes X

But for quantum computation we need programminglanguages that apply to noncartesian categories mdashbecause you canrsquot duplicate quantum data

And in quantum computation using anyons therelevant categories are braided

For more detail read our paper in Bob Coeckersquos forth-coming book New Structures in Physics You can findit now You can find it now on the arXiv

Categories Physics Topology Logic Computationobject system manifold proposition data type

morphism process cobordism proof program

zzzzzzz||

` `

` `

XampY ` Z λxλy x(y)

In general relativity objects in nCob describe choicesof space while morphisms describe choices of space-time I believe that

Quantum theory will eventually make more sense aspart of a theory of quantum gravity mdash but this can onlybe understood using categories

Why The weird features of quantum theory comefrom the ways that Hilb is less like Set than nCobBut nCob is what we use to describe space and space-time in general relativity

lsquoWeirdrsquo properties of quantum theory correspond tounsurprising properties of spacetime

object morphismbull bull rarr bull

SET set function betweenTHEORY sets

QUANTUM Hilbert space operator betweenTHEORY Hilbert spaces

(state) (process)

GENERAL manifold cobordism betweenRELATIVITY manifolds

(space) (spacetime)

For example Set is lsquocartesianrsquo while nCob and Hilbare not

If a symmetric monoidal category is cartesian youcan do various things including duplication

∆X X rarr X otimes X

In Set we can duplicate as follows

∆X X rarr X times Xx 7rarr (x x)

In Hilb we cannot duplicate the function

X rarr X otimes Xx 7rarr x otimes x

is not linear Itrsquos not a morphism in HilbSo we lsquocannot clone a quantum statersquo

Similarly in nCob there is no duplication despitethis misleading picture for n = 2

X

X otimes X

When n = 1 therersquos typically no cobordism from amanifold X to X otimes X and similarly for n = 4

What about logic and computer science These toostudy categories of things and processes

In proof theory we use categories where

bull an object is a proposition

bull a morphism is a proof

In computer science we use categories where

bull an object is a data type

bull a morphism is a program

In proof theory X ` Y means assuming X we canprove Y But we can also let it mean the set of proofsleading from assumption X to conclusion Y

Since proofs are morphisms we can compose themX ` Y Y ` Z

X ` Z

The identity morphism

X ` X

Logic uses monoidal categories where the tensorproduct is lsquoandrsquo We can tensor propositions andtensor proofs

W ` X Y ` ZWampY ` XampZ

In fact logic uses symmetric monoidal categoriesX ` YampZX ` ZampY

Classical logic is cartesian so it permits duplicationX ` Y

X ` YampY

Linear logic does not

A program that takes data of type X as input andreturns data of type Y can be seen as a morphismf X rarr Y

Categories of data types and programs are monoidalGiven data types X and Xprime there is a data type XotimesXprimeAnd given programs f X rarr Y f Xprime rarr Yprime we canwrite a program f otimes f prime that does these two jobs inparallel

fX

Yf prime

Xprime

Yprime= f otimes f prime

X otimes Xprime

Y otimes Yprime

These categories are are typically symmetric monoidal

X Y

Theyrsquore also cartesian For example we can writeprograms that duplicate data

∆X X rarr X otimes X

But for quantum computation we need programminglanguages that apply to noncartesian categories mdashbecause you canrsquot duplicate quantum data

And in quantum computation using anyons therelevant categories are braided

For more detail read our paper in Bob Coeckersquos forth-coming book New Structures in Physics You can findit now You can find it now on the arXiv

Categories Physics Topology Logic Computationobject system manifold proposition data type

morphism process cobordism proof program

zzzzzzz||

` `

` `

XampY ` Z λxλy x(y)

object morphismbull bull rarr bull

SET set function betweenTHEORY sets

QUANTUM Hilbert space operator betweenTHEORY Hilbert spaces

(state) (process)

GENERAL manifold cobordism betweenRELATIVITY manifolds

(space) (spacetime)

For example Set is lsquocartesianrsquo while nCob and Hilbare not

If a symmetric monoidal category is cartesian youcan do various things including duplication

∆X X rarr X otimes X

In Set we can duplicate as follows

∆X X rarr X times Xx 7rarr (x x)

In Hilb we cannot duplicate the function

X rarr X otimes Xx 7rarr x otimes x

is not linear Itrsquos not a morphism in HilbSo we lsquocannot clone a quantum statersquo

Similarly in nCob there is no duplication despitethis misleading picture for n = 2

X

X otimes X

When n = 1 therersquos typically no cobordism from amanifold X to X otimes X and similarly for n = 4

What about logic and computer science These toostudy categories of things and processes

In proof theory we use categories where

bull an object is a proposition

bull a morphism is a proof

In computer science we use categories where

bull an object is a data type

bull a morphism is a program

In proof theory X ` Y means assuming X we canprove Y But we can also let it mean the set of proofsleading from assumption X to conclusion Y

Since proofs are morphisms we can compose themX ` Y Y ` Z

X ` Z

The identity morphism

X ` X

Logic uses monoidal categories where the tensorproduct is lsquoandrsquo We can tensor propositions andtensor proofs

W ` X Y ` ZWampY ` XampZ

In fact logic uses symmetric monoidal categoriesX ` YampZX ` ZampY

Classical logic is cartesian so it permits duplicationX ` Y

X ` YampY

Linear logic does not

A program that takes data of type X as input andreturns data of type Y can be seen as a morphismf X rarr Y

Categories of data types and programs are monoidalGiven data types X and Xprime there is a data type XotimesXprimeAnd given programs f X rarr Y f Xprime rarr Yprime we canwrite a program f otimes f prime that does these two jobs inparallel

fX

Yf prime

Xprime

Yprime= f otimes f prime

X otimes Xprime

Y otimes Yprime

These categories are are typically symmetric monoidal

X Y

Theyrsquore also cartesian For example we can writeprograms that duplicate data

∆X X rarr X otimes X

But for quantum computation we need programminglanguages that apply to noncartesian categories mdashbecause you canrsquot duplicate quantum data

And in quantum computation using anyons therelevant categories are braided

For more detail read our paper in Bob Coeckersquos forth-coming book New Structures in Physics You can findit now You can find it now on the arXiv

Categories Physics Topology Logic Computationobject system manifold proposition data type

morphism process cobordism proof program

zzzzzzz||

` `

` `

XampY ` Z λxλy x(y)

For example Set is lsquocartesianrsquo while nCob and Hilbare not

If a symmetric monoidal category is cartesian youcan do various things including duplication

∆X X rarr X otimes X

In Set we can duplicate as follows

∆X X rarr X times Xx 7rarr (x x)

In Hilb we cannot duplicate the function

X rarr X otimes Xx 7rarr x otimes x

is not linear Itrsquos not a morphism in HilbSo we lsquocannot clone a quantum statersquo

Similarly in nCob there is no duplication despitethis misleading picture for n = 2

X

X otimes X

When n = 1 therersquos typically no cobordism from amanifold X to X otimes X and similarly for n = 4

What about logic and computer science These toostudy categories of things and processes

In proof theory we use categories where

bull an object is a proposition

bull a morphism is a proof

In computer science we use categories where

bull an object is a data type

bull a morphism is a program

In proof theory X ` Y means assuming X we canprove Y But we can also let it mean the set of proofsleading from assumption X to conclusion Y

Since proofs are morphisms we can compose themX ` Y Y ` Z

X ` Z

The identity morphism

X ` X

Logic uses monoidal categories where the tensorproduct is lsquoandrsquo We can tensor propositions andtensor proofs

W ` X Y ` ZWampY ` XampZ

In fact logic uses symmetric monoidal categoriesX ` YampZX ` ZampY

Classical logic is cartesian so it permits duplicationX ` Y

X ` YampY

Linear logic does not

A program that takes data of type X as input andreturns data of type Y can be seen as a morphismf X rarr Y

Categories of data types and programs are monoidalGiven data types X and Xprime there is a data type XotimesXprimeAnd given programs f X rarr Y f Xprime rarr Yprime we canwrite a program f otimes f prime that does these two jobs inparallel

fX

Yf prime

Xprime

Yprime= f otimes f prime

X otimes Xprime

Y otimes Yprime

These categories are are typically symmetric monoidal

X Y

Theyrsquore also cartesian For example we can writeprograms that duplicate data

∆X X rarr X otimes X

But for quantum computation we need programminglanguages that apply to noncartesian categories mdashbecause you canrsquot duplicate quantum data

And in quantum computation using anyons therelevant categories are braided

For more detail read our paper in Bob Coeckersquos forth-coming book New Structures in Physics You can findit now You can find it now on the arXiv

Categories Physics Topology Logic Computationobject system manifold proposition data type

morphism process cobordism proof program

zzzzzzz||

` `

` `

XampY ` Z λxλy x(y)

In Hilb we cannot duplicate the function

X rarr X otimes Xx 7rarr x otimes x

is not linear Itrsquos not a morphism in HilbSo we lsquocannot clone a quantum statersquo

Similarly in nCob there is no duplication despitethis misleading picture for n = 2

X

X otimes X

When n = 1 therersquos typically no cobordism from amanifold X to X otimes X and similarly for n = 4

What about logic and computer science These toostudy categories of things and processes

In proof theory we use categories where

bull an object is a proposition

bull a morphism is a proof

In computer science we use categories where

bull an object is a data type

bull a morphism is a program

In proof theory X ` Y means assuming X we canprove Y But we can also let it mean the set of proofsleading from assumption X to conclusion Y

Since proofs are morphisms we can compose themX ` Y Y ` Z

X ` Z

The identity morphism

X ` X

Logic uses monoidal categories where the tensorproduct is lsquoandrsquo We can tensor propositions andtensor proofs

W ` X Y ` ZWampY ` XampZ

In fact logic uses symmetric monoidal categoriesX ` YampZX ` ZampY

Classical logic is cartesian so it permits duplicationX ` Y

X ` YampY

Linear logic does not

A program that takes data of type X as input andreturns data of type Y can be seen as a morphismf X rarr Y

Categories of data types and programs are monoidalGiven data types X and Xprime there is a data type XotimesXprimeAnd given programs f X rarr Y f Xprime rarr Yprime we canwrite a program f otimes f prime that does these two jobs inparallel

fX

Yf prime

Xprime

Yprime= f otimes f prime

X otimes Xprime

Y otimes Yprime

These categories are are typically symmetric monoidal

X Y

Theyrsquore also cartesian For example we can writeprograms that duplicate data

∆X X rarr X otimes X

But for quantum computation we need programminglanguages that apply to noncartesian categories mdashbecause you canrsquot duplicate quantum data

And in quantum computation using anyons therelevant categories are braided

For more detail read our paper in Bob Coeckersquos forth-coming book New Structures in Physics You can findit now You can find it now on the arXiv

Categories Physics Topology Logic Computationobject system manifold proposition data type

morphism process cobordism proof program

zzzzzzz||

` `

` `

XampY ` Z λxλy x(y)

What about logic and computer science These toostudy categories of things and processes

In proof theory we use categories where

bull an object is a proposition

bull a morphism is a proof

In computer science we use categories where

bull an object is a data type

bull a morphism is a program

In proof theory X ` Y means assuming X we canprove Y But we can also let it mean the set of proofsleading from assumption X to conclusion Y

Since proofs are morphisms we can compose themX ` Y Y ` Z

X ` Z

The identity morphism

X ` X

Logic uses monoidal categories where the tensorproduct is lsquoandrsquo We can tensor propositions andtensor proofs

W ` X Y ` ZWampY ` XampZ

In fact logic uses symmetric monoidal categoriesX ` YampZX ` ZampY

Classical logic is cartesian so it permits duplicationX ` Y

X ` YampY

Linear logic does not

A program that takes data of type X as input andreturns data of type Y can be seen as a morphismf X rarr Y

Categories of data types and programs are monoidalGiven data types X and Xprime there is a data type XotimesXprimeAnd given programs f X rarr Y f Xprime rarr Yprime we canwrite a program f otimes f prime that does these two jobs inparallel

fX

Yf prime

Xprime

Yprime= f otimes f prime

X otimes Xprime

Y otimes Yprime

These categories are are typically symmetric monoidal

X Y

Theyrsquore also cartesian For example we can writeprograms that duplicate data

∆X X rarr X otimes X

But for quantum computation we need programminglanguages that apply to noncartesian categories mdashbecause you canrsquot duplicate quantum data

And in quantum computation using anyons therelevant categories are braided

For more detail read our paper in Bob Coeckersquos forth-coming book New Structures in Physics You can findit now You can find it now on the arXiv

Categories Physics Topology Logic Computationobject system manifold proposition data type

morphism process cobordism proof program

zzzzzzz||

` `

` `

XampY ` Z λxλy x(y)

In proof theory X ` Y means assuming X we canprove Y But we can also let it mean the set of proofsleading from assumption X to conclusion Y

Since proofs are morphisms we can compose themX ` Y Y ` Z

X ` Z

The identity morphism

X ` X

Logic uses monoidal categories where the tensorproduct is lsquoandrsquo We can tensor propositions andtensor proofs

W ` X Y ` ZWampY ` XampZ

In fact logic uses symmetric monoidal categoriesX ` YampZX ` ZampY

Classical logic is cartesian so it permits duplicationX ` Y

X ` YampY

Linear logic does not

A program that takes data of type X as input andreturns data of type Y can be seen as a morphismf X rarr Y

Categories of data types and programs are monoidalGiven data types X and Xprime there is a data type XotimesXprimeAnd given programs f X rarr Y f Xprime rarr Yprime we canwrite a program f otimes f prime that does these two jobs inparallel

fX

Yf prime

Xprime

Yprime= f otimes f prime

X otimes Xprime

Y otimes Yprime

These categories are are typically symmetric monoidal

X Y

Theyrsquore also cartesian For example we can writeprograms that duplicate data

∆X X rarr X otimes X

But for quantum computation we need programminglanguages that apply to noncartesian categories mdashbecause you canrsquot duplicate quantum data

And in quantum computation using anyons therelevant categories are braided

For more detail read our paper in Bob Coeckersquos forth-coming book New Structures in Physics You can findit now You can find it now on the arXiv

Categories Physics Topology Logic Computationobject system manifold proposition data type

morphism process cobordism proof program

zzzzzzz||

` `

` `

XampY ` Z λxλy x(y)

Logic uses monoidal categories where the tensorproduct is lsquoandrsquo We can tensor propositions andtensor proofs

W ` X Y ` ZWampY ` XampZ

In fact logic uses symmetric monoidal categoriesX ` YampZX ` ZampY

Classical logic is cartesian so it permits duplicationX ` Y

X ` YampY

Linear logic does not

A program that takes data of type X as input andreturns data of type Y can be seen as a morphismf X rarr Y

Categories of data types and programs are monoidalGiven data types X and Xprime there is a data type XotimesXprimeAnd given programs f X rarr Y f Xprime rarr Yprime we canwrite a program f otimes f prime that does these two jobs inparallel

fX

Yf prime

Xprime

Yprime= f otimes f prime

X otimes Xprime

Y otimes Yprime

These categories are are typically symmetric monoidal

X Y

Theyrsquore also cartesian For example we can writeprograms that duplicate data

∆X X rarr X otimes X

But for quantum computation we need programminglanguages that apply to noncartesian categories mdashbecause you canrsquot duplicate quantum data

And in quantum computation using anyons therelevant categories are braided

For more detail read our paper in Bob Coeckersquos forth-coming book New Structures in Physics You can findit now You can find it now on the arXiv

Categories Physics Topology Logic Computationobject system manifold proposition data type

morphism process cobordism proof program

zzzzzzz||

` `

` `

XampY ` Z λxλy x(y)

A program that takes data of type X as input andreturns data of type Y can be seen as a morphismf X rarr Y

Categories of data types and programs are monoidalGiven data types X and Xprime there is a data type XotimesXprimeAnd given programs f X rarr Y f Xprime rarr Yprime we canwrite a program f otimes f prime that does these two jobs inparallel

fX

Yf prime

Xprime

Yprime= f otimes f prime

X otimes Xprime

Y otimes Yprime

These categories are are typically symmetric monoidal

X Y

Theyrsquore also cartesian For example we can writeprograms that duplicate data

∆X X rarr X otimes X

But for quantum computation we need programminglanguages that apply to noncartesian categories mdashbecause you canrsquot duplicate quantum data

And in quantum computation using anyons therelevant categories are braided

For more detail read our paper in Bob Coeckersquos forth-coming book New Structures in Physics You can findit now You can find it now on the arXiv

Categories Physics Topology Logic Computationobject system manifold proposition data type

morphism process cobordism proof program

zzzzzzz||

` `

` `

XampY ` Z λxλy x(y)

These categories are are typically symmetric monoidal

X Y

Theyrsquore also cartesian For example we can writeprograms that duplicate data

∆X X rarr X otimes X

But for quantum computation we need programminglanguages that apply to noncartesian categories mdashbecause you canrsquot duplicate quantum data

And in quantum computation using anyons therelevant categories are braided

For more detail read our paper in Bob Coeckersquos forth-coming book New Structures in Physics You can findit now You can find it now on the arXiv

Categories Physics Topology Logic Computationobject system manifold proposition data type

morphism process cobordism proof program

zzzzzzz||

` `

` `

XampY ` Z λxλy x(y)

For more detail read our paper in Bob Coeckersquos forth-coming book New Structures in Physics You can findit now You can find it now on the arXiv

Categories Physics Topology Logic Computationobject system manifold proposition data type

morphism process cobordism proof program

zzzzzzz||

` `

` `

XampY ` Z λxλy x(y)