For setAI and selfAdjoint
That poll of yours is moving slowly, selfAdjoint. Just in case the informational option wins, I thought I would reintroduce it.
-----------------------------------------------------------------
CATEGORIES, GRAVITY, LOGIC AND THE COMPUTATIONAL UNIVERSE
-----------------------------------------------------------------
Recent interest in category theory amongst the String theorists, and the growing interest in the intersection between LQG and Strings, suggests that perhaps it is time to recognise the existence of QG ideas outside the scope of
Strings, Branes and LQG.
The ideas to which I refer do have an intersection with both Strings and LQG. In the first case, the notion of a gerbe, as discussed in
Higher Gauge Theory: 2-Connections on 2-Bundles John Baez,
Urs Schreiber,
http://www.arxiv.org/abs/hep-th/0412325
is a category theoretic one. In the second case, the spin foam (for a review see
Spin Foam Models for Quantum Gravity Alejandro Perez,
http://arxiv.org/abs/gr-qc/0301113 ) approach to QG uses the functorial aspect of topological field theories, and has its origins in Penrose's spin networks, which in turn arose from the study of twistors, about which more is said below.
This is a very short introduction to this subject. A few useful web references are collected. It is worth noting here
John Baez's homepage
http://math.ucr.edu/home/baez/README.html
---------------------------------
The Third Road to Quantum Gravity
---------------------------------
The third road is not about the application of a few category theoretic concepts, such as gerbes or functors, to physics modeled entirely on existing principles. It is about trying to understand what we mean by
observation and
quantum geometry at a fundamental level. The idea of a path integral summation over
preselected geometries is dismissed outright.
Only category theory can discuss logic, geometry, algebra and number theory in the same language. The third road says "get the logic right, and you'll see how computational the universe is".
Now this might all be pie-in-the-sky philosophy, but actually it is a well-developed approach to Quantum Gravity. The reason that it remains unrecognised as such is partly due to its interdisciplinary nature. Experts in logic tend to reside in Philosophy departments, experts in computation in Computer Science departments and so on.
For a real philosopher's introduction to these ideas see
Loop and Knots as topoi of substance R.E. Zimmermann
http://philsci-archive.pitt.edu/archive/00000385/00/0004077.pdf
or maybe look at some of my previous posts at
https://www.physicsforums.com/search.php?searchid=123330
In the next section, I would like to point out that General Relativity itself is category theoretic in nature.
--------------------------------
General Relativity and Categories
--------------------------------
Background independence is about more than coordinate invariance. I shouldn't have to say this, but String theorists don't seem to know this. If you take all the matter out of the universe then there
isn't any spacetime. Penrose understood this well. That is why he started using sheaves - to do twistor theory.
The question is: how can we describe a
point in spacetime? Well, a point in spacetime isn't of any physical importance. In fact it was only by realising this that Einstein came to accept general covariance in the first place (see the book by J. Stachel,
Einstein from B to Z Birkhauser 2002). What
is physical are the (equivalence classes of) gravitational fields.
If we work with sheaves over a space M then a point is indeed a highly derived concept. So the physics is telling us we should use sheaves to do GR. But sheaves are examples of functors - maps between categories.
Carrying this much further, one can model a differential manifold on something called a
local ringed topos, namely the topos of sheaves on some subset of \mathbb{R}^{n} which contains the distinguished sheaf of differentiable \mathbb{R} valued functions on the subset.
But why do we need manifolds at all? Some people take this question very seriously. See, for instance, the recent 400+ page tour-de-force
C^{\infty}-smooth singularities exposed: Chimeras of the differential spacetime manifold; A. Mallios, I. Raptis,
http://arxiv.org/abs/gr-qc/0411121
on the use of Abstract Differential Geometry in classical and quantum gravity, with its extensive bibliography.
Anyone still reading this will at least grudgingly admit that maybe a physicist needs to know a little bit about what a category is...
--------------------------------
Quick Introduction to Categories
--------------------------------
Whereas a set has elements, and a map between sets takes elements to elements, a category has both elements, called
objects, and relationships between elements, called
arrows. Every object A is equipped with at least an identity arrow 1_{A} from A to A. Maps between categories, called
functors, take objects to objects and arrows to arrows. Arrows may be composed f \circ g if their ends match appropriately. An arrow is
monic if for any g: A \rightarrow B and h: A \rightarrow B, f \circ g = f \circ h implies g = h.
For example, there is a category \mathbf{Set} whose objects are sets and whose arrows are functions between sets. In \mathbf{Set} there is an object \{ 0,1 \}. There are also many arrows of the form f: S \rightarrow \{ 0,1 \} for a set S. Such arrows may be thought of as the selection of a subset of
S, namely those elements that are mapped to 1. A one element set, \{ \ast \}, has precisely one arrow into it from any other set, making it an example of a
terminal object in \mathbf{Set}.
Functors are contravariant if they actually act on the category with all arrows reversed. Contravariant functors from a (small) category C into \mathbf{Set} are known as
presheaves, providing a preliminary example of a topos. When C comes equipped with a topology (definition omitted) one restricts to a subcategory of
sheaves.
The intended interpretation of pieces of categories is that they are geometric entities. Objects are zero dimensional and arrows are one dimensional. In a category there is no equality between objects, but we consider objects isomorphic if there exists two arrows f and g such that f \circ g = 1_{A} and g \circ f = 1_{B}.
Now one may also consider the category \mathbf{Cat}, with categories as objects (which are small enough in a suitable sense) and arrows functors between them. One may naturally include in this category the
natural transformations \tau between functors, as another level of arrows, as some commuting squares, which I would like to draw but I need xypic...These squares may be composed, both vertically and horizontally, in the obvious way. Thus \mathbf{Cat} is an example of a 2-category: an inherently two dimensional structure. In a 2-category, all arrows between two objects A and B, denoted Hom(A,B), form a category.
Another example of a 2-category is the category of topological spaces, with homeomorphisms for 1-arrows and homotopy maps as 2-arrows.
Given a subset S of the arrows of a category C one defines the localisation category S^{-1} (C) by sending all arrows in S to isomorphisms under a functor C \rightarrow S^{-1}(C) which has a nice universal property.
A category representing the ordinal \mathbf{4} is visualised as a 3-simplex equipped with oriented edges and faces.
Recall that in three dimensions gravity is a topological theory because it has no local degrees of freedom. If one is interested in (physical) spaces that are topological (ie. there is an equivalence up to continuous deformation) and oriented it is sufficient to describe them by a space made out of simplices,
suitably glued together. A TFT is, axiomatically, a functor from such spaces, thought of as arrows between boundary components, into an algebraic category.
However, this isn't category-theoretic enough for the third road.
