chramostak> zrovna to ctu :)
Before we begin our chronology, to help the reader keep from getting lost in a cloud of details,
it will be helpful to sketch the road ahead. Why did categories turn out to be useful in physics?
The reason is ultimately very simple. A category consists of `objects' x; y; z; : : : and `morphisms'
which go between objects, for example
f : x -> y
A good example is the category of Hilbert spaces, where the objects are Hilbert spaces and the
morphisms are bounded operators. In physics we can think of an object as a `state space' for
some physical system, and a morphism as a `process' taking states of one system to states of
another (perhaps the same one). In short, we use objects to describe kinematics, and morphisms
to describe dynamics.
This escalation of dimensions can continue. In the diagrams Feynman used to describe interacting
particles, we can continuously interpolate between this way of switching two particles.
This requires four dimensions: one of time and three of space. To formalize this algebraically we
need a `symmetric monoidal category', which is a special sort of 4-category.
More general n-categories, including those for higher values of n, may also be useful in physics.
This is especially true in string theory and spin foam models of quantum gravity. These theories
describe strings, graphs, and their higher-dimensional generalizations propagating in spacetimes
which may themselves have more than 4 dimensions.
So, in abstract the idea is simple: we can use n-categories to algebraically formalize physical
theories in which processes can be depicted geometrically using n-dimensional diagrams.