Notes on: Baez, J. C., & Stay, M. (2009): Physics, topology, logic and computation: a rosetta stone
Table of Contents
Progress
- Note taken on
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 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
- Tang_{k}: 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
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 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:
- for all the traingle equation holds baez09_physic_topol_logic_comput
- for all , 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 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 .
- [baez09_physic_topol_logic_comput] Baez & Stay, Physics, Topology, Logic and Computation: A Rosetta Stone, CoRR, (2009). link.