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

## Progress

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

## 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 is a n-dimensional manifold whose boundary is the disjoint union of the (n - 1)-dimensional manifolds and . baez09_physic_topol_logic_comput

The boundary of an dimensional manifold is an n-dimensional manifold 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 has a finite subcover.

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

there exists a finite subset of s.t.

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 consists of:

• a collection of objects, where if is an object of we wrte
• for every pair of objects , a set of morphisms from to . We call this set a homset. If , then we write

such that:

• for every object there is an identity morphism
• morphisms are composable: given and , there is a composite morphism denoted
• an identiy morphism is both a left and right unit for composition: if , then
• composition is associative:

We say a morphism is an isomorphism if it has an inverse; that is, there exists another morphism such that and .

#### 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 of categories and is the category where:

• an object is a consisting of an object and an object
• a morphism from to is a pair consisting of a morphism and a morphism
• composition is done componentwise:
• identity morphisms are defined componentwise:

A monoidal category consists of:

• a category
• a tensor product functor :
• a unit object
• a natural isomorphism called the associator, assigning to each triple of objects an isomorphism

• natural isomorphisms called the left and right unitors, assigning to each object isomorphisms

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 to .

#### Cartesian product

Given objects and in some category, we say an object is equipped with morphisms

is a cartesian product (or simply product ) of and if for any object and morphisms See paper there exists a unique morphism making the following diagram commute: See paper (That is, and .) 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 and when it exists, and denoting it as .

#### Terminal object

An object in a category is terminal if for any object there exists a unique morphism from to , which we denote as .

### Functors

A functior from a category to a category is a map sending:

• any object to an object
• any morphism in to a morphism in

in such a way that:

• preserves identities: for any object
• preserves composition: for any pair of morphism in ,

It might be useful to think of a functor as a providing a representation of objects in some category in the category .

#### Nautral transformation

Given two functors , a natural transformation assigns to every object a morphism such that for any morphism in , the equation holds in .

A natural isomorphism between functors is a natural transformation such that is an isomorphism for every .

## 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 .