Notes on: Baez, J. C., & Stay, M. (2009): Physics, topology, logic and computation: a rosetta stone

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- TODO Finish note about paranthesizing tensor products
Notation
Definitions
General
- Topology and Quantum physics

Progress

Note taken on [2017-10-16 Mon 09:14]
p. 13

TODO Finish note about paranthesizing tensor products

Notation

Definitions

Cobordism is a fundamenetal equivalence relation on the class of compact manifolds of the same dimension, set up using the concept of the "boundary" of a manifold. Two manifolds are of the same dimension are cobordant if their disjoin union is the boundary of a compact manifold one dimension higher.

A cobordism $baez09_physic_topol_logic_comput_1c514dff8af4495fd39e5bc92e5fa4386dcede0f.png$ is a n-dimensional manifold whose boundary is the disjoint union of the (n - 1)-dimensional manifolds $baez09_physic_topol_logic_comput_0207be880056b9a69e22e729dd37bced29cd174a.png$ and $baez09_physic_topol_logic_comput_3ca302b80fb078e124ac6b194794815069394f1a.png$ . baez09_physic_topol_logic_comput

The boundary of an $baez09_physic_topol_logic_comput_8c156adfcaf78381ea781a88c6434e0398eaae92.png$ dimensional manifold $baez09_physic_topol_logic_comput_f75d7681f9d0acecef24eb637ee201aa5b39199a.png$ is an n-dimensional manifold $baez09_physic_topol_logic_comput_eaa8282f030f36d1eca31e9dd5bf0f2e9067036d.png$ that is closed , i.e. with empty boundary.

Words

Set: the category where objects are sets
Hilb: the category where objects are finite-dimensional Hilbert spaces
nCob: the category where morphisms are n-dimensional cobordisms
Tang_k: the category where morphisms are k-codimensional tangles

Compact space

A topological space is compact if every open cover of $baez09_physic_topol_logic_comput_0207be880056b9a69e22e729dd37bced29cd174a.png$ has a finite subcover.

In other words, if $baez09_physic_topol_logic_comput_0207be880056b9a69e22e729dd37bced29cd174a.png$ is a union of a family of open sets, there is a finite subfamily whose union is $baez09_physic_topol_logic_comput_0207be880056b9a69e22e729dd37bced29cd174a.png$ , or equivalently $baez09_physic_topol_logic_comput_0207be880056b9a69e22e729dd37bced29cd174a.png$ is compact if for every collection $baez09_physic_topol_logic_comput_e20dcda5f035650122343c61053a7c3ad6acacaa.png$ of open subsets of $baez09_physic_topol_logic_comput_0207be880056b9a69e22e729dd37bced29cd174a.png$ such that

$baez09_physic_topol_logic_comput_b63a9478805b8525a4bfb594ffcbb7eb105a820f.png$

there exists a finite subset $baez09_physic_topol_logic_comput_e08421ac16591ee60adecc1e14a359550f81a1e2.png$ of $baez09_physic_topol_logic_comput_e20dcda5f035650122343c61053a7c3ad6acacaa.png$ s.t.

$baez09_physic_topol_logic_comput_444f59c52d57c48638cd41ebd7c5fb1f9aab70d7.png$

It's a generalization of closed and bounded sets in Euclidean space.

For topology, one can also think of a compact manifold as a "manifold without boundary".

Categories

A category $baez09_physic_topol_logic_comput_e20dcda5f035650122343c61053a7c3ad6acacaa.png$ consists of:

a collection of objects, where if $baez09_physic_topol_logic_comput_0207be880056b9a69e22e729dd37bced29cd174a.png$ is an object of $baez09_physic_topol_logic_comput_e20dcda5f035650122343c61053a7c3ad6acacaa.png$ we wrte $baez09_physic_topol_logic_comput_99489886f57a186a2ebac5f4d117d2331fa9a856.png$
for every pair of objects $baez09_physic_topol_logic_comput_3a2e2752cac74a1a51b468d725a7cfaf3ff49621.png$ , a set $baez09_physic_topol_logic_comput_040222febefe2b406119169efa71d5c0d4c11001.png$ of morphisms from $baez09_physic_topol_logic_comput_0207be880056b9a69e22e729dd37bced29cd174a.png$ to $baez09_physic_topol_logic_comput_3ca302b80fb078e124ac6b194794815069394f1a.png$ . We call this set $baez09_physic_topol_logic_comput_040222febefe2b406119169efa71d5c0d4c11001.png$ a homset. If $baez09_physic_topol_logic_comput_ef724430adffbf61ddf00089d5d9e23405247c4f.png$ , then we write $baez09_physic_topol_logic_comput_9f41124d9c63e1e1d122e998ced5368666463046.png$

such that:

for every object $baez09_physic_topol_logic_comput_0207be880056b9a69e22e729dd37bced29cd174a.png$ there is an identity morphism $baez09_physic_topol_logic_comput_bb42f15315b7c0fe7b82a07fb7b86d27e01cc558.png$
morphisms are composable: given $baez09_physic_topol_logic_comput_1c514dff8af4495fd39e5bc92e5fa4386dcede0f.png$ and $baez09_physic_topol_logic_comput_8a5a8f381711294949ed45ca9fe4f884abb434dc.png$ , there is a composite morphism $baez09_physic_topol_logic_comput_ec3ef598b57495932e8f76d5ff6c5048fb247937.png$ denoted $baez09_physic_topol_logic_comput_e8649f4a12a05a97316f4696660566092638507f.png$
an identiy morphism is both a left and right unit for composition: if $baez09_physic_topol_logic_comput_8333ad5a177802c4817004efe6fb904ffccdf12f.png$ , then $baez09_physic_topol_logic_comput_32b0a55219033f9d171bd466ea1e250988838545.png$
composition is associative: $baez09_physic_topol_logic_comput_9319d9318fb94f59bd53480ba5f4d80549aacd4b.png$

We say a morphism $baez09_physic_topol_logic_comput_1c514dff8af4495fd39e5bc92e5fa4386dcede0f.png$ is an isomorphism if it has an inverse; that is, there exists another morphism $baez09_physic_topol_logic_comput_4b9b7b55e630a71b704661364d571a8ca8dacd4b.png$ such that $baez09_physic_topol_logic_comput_36578371229e7dce947c5003c1d76cd842660349.png$ and $baez09_physic_topol_logic_comput_da8582a741479847757fa309c72b12e9d55474dd.png$ .

Specific categories

nCob

Objects are (n - 1)-dimensional manifolds and the morphisms are n-dimensional cobordisms

Tank_k

collection of points in k-dimensional cube
morphism is a 'tangle': a collection of arcs and circles smoothly embedded in a (k + 1)-dimensional cube, s.t. circles lie in the interior of the cube, while the arcs touch the boundary of the cube only at its top and bottom, only at their endpoints.

A bit more precisly, tangles are 'isotopy classes' of such embedded arcs and circles: this equivalence relation means that only the topology of the tangle matters, not its geometry .

Monoidal categories

The cartesian product $baez09_physic_topol_logic_comput_f06f4716a4bd13891dc445f292767e67c9283228.png$ of categories $baez09_physic_topol_logic_comput_e20dcda5f035650122343c61053a7c3ad6acacaa.png$ and $baez09_physic_topol_logic_comput_0986d7658d3053612827f5f4167893f0a2261610.png$ is the category where:

an object is a $baez09_physic_topol_logic_comput_81cc4bae4604f275a63afc2677c6aa9f2fc98196.png$ consisting of an object $baez09_physic_topol_logic_comput_99489886f57a186a2ebac5f4d117d2331fa9a856.png$ and an object $baez09_physic_topol_logic_comput_99c26366a339af328a2814629af6fb96ee824297.png$
a morphism from $baez09_physic_topol_logic_comput_d292723aef6a54a3fa5357a5bc33b24d57dcda03.png$ to $baez09_physic_topol_logic_comput_9065eb2ea9431c134ed3a7f8434f6a336d92f605.png$ is a pair $baez09_physic_topol_logic_comput_111a81ddcce47c1aed5abbed3801150e0c4c61d2.png$ consisting of a morphism $baez09_physic_topol_logic_comput_1c514dff8af4495fd39e5bc92e5fa4386dcede0f.png$ and a morphism $baez09_physic_topol_logic_comput_2ad81a642e4c950445ca671fb73b8064702a6ba3.png$
composition is done componentwise: $baez09_physic_topol_logic_comput_fd29b097a99a0deba990c21936f797d6e69d9262.png$
identity morphisms are defined componentwise: $baez09_physic_topol_logic_comput_e7d7f3bd450235fcb4ca7eb572bc7f5b73bfc47d.png$

A monoidal category consists of:

a category $baez09_physic_topol_logic_comput_e20dcda5f035650122343c61053a7c3ad6acacaa.png$
a tensor product functor $baez09_physic_topol_logic_comput_7f953d3d4880831f968bdd52594ee674f1c09511.png$ : $baez09_physic_topol_logic_comput_0733314c5eb0f2ff8d2f7dc8a3de68e874aca575.png$
a unit object $baez09_physic_topol_logic_comput_287554475261d4779b148f01795a367599dfab02.png$
a natural isomorphism called the associator, assigning to each triple of objects $baez09_physic_topol_logic_comput_0600f35b0761fd87b03bc09b40f4c0e6e8820a41.png$ an isomorphism

$baez09_physic_topol_logic_comput_3399980d049fc92e5045e458f133940ed1c30159.png$
natural isomorphisms called the left and right unitors, assigning to each object $baez09_physic_topol_logic_comput_99489886f57a186a2ebac5f4d117d2331fa9a856.png$ isomorphisms

$baez09_physic_topol_logic_comput_0bc19eb424843115f12a391d3cfeeef2f99694e5.png$

such that:

for all $baez09_physic_topol_logic_comput_d059caaf56756f295d477cd84d48553bc9773c36.png$ the traingle equation holds baez09_physic_topol_logic_comput
for all $baez09_physic_topol_logic_comput_96fed14ebe360a0161165c8c38601afc7bd5c601.png$ , the pentagon equation holds baez09_physic_topol_logic_comput

A tensor product of four objects has five ways to paranthesize it, and at first glance the associator lets us build two isomorphisms from $baez09_physic_topol_logic_comput_cee029cec7349f1c6386213cea5649a2e64017df.png$ to $baez09_physic_topol_logic_comput_32e188bb312501ad990bde03788edf271c79aa4b.png$ .

Cartesian product

Given objects $baez09_physic_topol_logic_comput_0207be880056b9a69e22e729dd37bced29cd174a.png$ and $baez09_physic_topol_logic_comput_8b60383d4f6295519c0485ce200161b0bf77009f.png$ in some category, we say an object $baez09_physic_topol_logic_comput_432891abb4f9ff81fc2eff6964c1922796fd50b9.png$ is equipped with morphisms

$baez09_physic_topol_logic_comput_a5dfd7b7d777093be5538f1ab1d7b5627f138fd6.png$

is a cartesian product (or simply product ) of $baez09_physic_topol_logic_comput_0207be880056b9a69e22e729dd37bced29cd174a.png$ and $baez09_physic_topol_logic_comput_8b60383d4f6295519c0485ce200161b0bf77009f.png$ if for any object $baez09_physic_topol_logic_comput_88c6e1c008509ea8de35316286b272c4107bed01.png$ and morphisms See paper there exists a unique morphism $baez09_physic_topol_logic_comput_dc310c6b438cb2ce44d5542c477c07081bffe5d7.png$ making the following diagram commute: See paper (That is, $baez09_physic_topol_logic_comput_2017b9e96719dd69808d0a5a07b5e42b0719fdb4.png$ and $baez09_physic_topol_logic_comput_723c225e3e10c0ee17682f58d8e9ac139e74ce01.png$ .) We say a category has binary products if every pair of objects has a product.

This product may not exist, and it may not be unique, but when it does exists it is unique up to a canonical isomorphism. Therefore we talk about the product of objects $baez09_physic_topol_logic_comput_0207be880056b9a69e22e729dd37bced29cd174a.png$ and $baez09_physic_topol_logic_comput_3ca302b80fb078e124ac6b194794815069394f1a.png$ when it exists, and denoting it as $baez09_physic_topol_logic_comput_380c396368fabb7e42f3a29df3bdab8f309d0b6f.png$ .

Terminal object

An object $baez09_physic_topol_logic_comput_7ba43ff00864c097bf5a587573e23164e2417a0b.png$ in a category $baez09_physic_topol_logic_comput_e20dcda5f035650122343c61053a7c3ad6acacaa.png$ is terminal if for any object $baez09_physic_topol_logic_comput_0d49c152dcd0c4ae89f56612beddb2c6683ffb14.png$ there exists a unique morphism from $baez09_physic_topol_logic_comput_88c6e1c008509ea8de35316286b272c4107bed01.png$ to $baez09_physic_topol_logic_comput_7ba43ff00864c097bf5a587573e23164e2417a0b.png$ , which we denote as $baez09_physic_topol_logic_comput_a7bf3de973b8a63d78bf2f679bb6230558d2157b.png$ .

Functors

A functior $baez09_physic_topol_logic_comput_29273746ef90faa7674d07d1bd16294875017104.png$ from a category $baez09_physic_topol_logic_comput_e20dcda5f035650122343c61053a7c3ad6acacaa.png$ to a category $baez09_physic_topol_logic_comput_b689cba8d7566f6adaf605a844e193a27e155078.png$ is a map sending:

any object $baez09_physic_topol_logic_comput_99489886f57a186a2ebac5f4d117d2331fa9a856.png$ to an object $baez09_physic_topol_logic_comput_935c70f95331f13f7768051c94272e1bd0bfcd00.png$
any morphism $baez09_physic_topol_logic_comput_1c514dff8af4495fd39e5bc92e5fa4386dcede0f.png$ in $baez09_physic_topol_logic_comput_e20dcda5f035650122343c61053a7c3ad6acacaa.png$ to a morphism $baez09_physic_topol_logic_comput_f68cecb23d624d66b6cfe0cad6435fdcd8e5e2dd.png$ in $baez09_physic_topol_logic_comput_b689cba8d7566f6adaf605a844e193a27e155078.png$

in such a way that:

$baez09_physic_topol_logic_comput_e08421ac16591ee60adecc1e14a359550f81a1e2.png$ preserves identities: for any object $baez09_physic_topol_logic_comput_795ec46d05dd626b3586db12cf33e21b4abc4d25.png$
$baez09_physic_topol_logic_comput_e08421ac16591ee60adecc1e14a359550f81a1e2.png$ preserves composition: for any pair of morphism $baez09_physic_topol_logic_comput_484f423d9d0c852ec091c876135cb87f86464fa8.png$ in $baez09_physic_topol_logic_comput_e20dcda5f035650122343c61053a7c3ad6acacaa.png$ , $baez09_physic_topol_logic_comput_aedae54fef6746dd7156c799b30b0f9f7e14768b.png$

It might be useful to think of a functor $baez09_physic_topol_logic_comput_29273746ef90faa7674d07d1bd16294875017104.png$ as a providing a representation of objects in some category $baez09_physic_topol_logic_comput_e20dcda5f035650122343c61053a7c3ad6acacaa.png$ in the category $baez09_physic_topol_logic_comput_b689cba8d7566f6adaf605a844e193a27e155078.png$ .

Nautral transformation

Given two functors $baez09_physic_topol_logic_comput_cb026a49cb7460ace4bf42840381590df20b2da2.png$ , a natural transformation $baez09_physic_topol_logic_comput_77930595fbeb8bfcb819c605f128401df24caee5.png$ assigns to every object $baez09_physic_topol_logic_comput_99489886f57a186a2ebac5f4d117d2331fa9a856.png$ a morphism $baez09_physic_topol_logic_comput_75998bb3d3f6730228195b3948ca2ef5dc394c84.png$ such that for any morphism $baez09_physic_topol_logic_comput_1c514dff8af4495fd39e5bc92e5fa4386dcede0f.png$ in $baez09_physic_topol_logic_comput_e20dcda5f035650122343c61053a7c3ad6acacaa.png$ , the equation $baez09_physic_topol_logic_comput_a877476bcb83b9098374d1979d5544ebf6f084b2.png$ holds in $baez09_physic_topol_logic_comput_b689cba8d7566f6adaf605a844e193a27e155078.png$ .

See baez09_physic_topol_logic_comput.

A natural isomorphism between functors $baez09_physic_topol_logic_comput_cb026a49cb7460ace4bf42840381590df20b2da2.png$ is a natural transformation $baez09_physic_topol_logic_comput_77930595fbeb8bfcb819c605f128401df24caee5.png$ such that $baez09_physic_topol_logic_comput_568d698add7c7ea46df24bc9e72fadcaefb5956b.png$ is an isomorphism for every $baez09_physic_topol_logic_comput_99489886f57a186a2ebac5f4d117d2331fa9a856.png$ .

General

Topology and Quantum physics

In quantum physics we use categories where objects are Hilbert spaces and morphisms are bounded linear operators . We specify a system by giving a Hilbert space, but this Hilbert space is not really the set of states of the system: a state is actually a ray in the Hilbert space. Similarily, the bounded linear operator is not precisely a function from states of one system to states of another.

Another category in physics has objects representing collections of particles, and morphisms representing their worldlines and interactions .

[baez09_physic_topol_logic_comput] Baez & Stay, Physics, Topology, Logic and Computation: A Rosetta Stone, CoRR, (2009). link.