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

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  • 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
  • Tangk: 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
Tankk
  • 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:

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 .