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Category nCob in General Relativity
As was shown in [1], it is possible to describe General Relativity in terms of the Category nCob, in which we have the following:- Objects are identified with (arbitrary) (n-1)-dimensional manifolds which represent space at a given time.
- Morphisms are identified with n-dimensional manifolds which represent spacetime (also called cobordism). The conditions on this cobordism are such that given, two (n-1)-manifolds S and , then, M is a cobordism between S and iff the boundary of M is the union of S and . It is useful to think of M as a process which changes the Topological structure of space, i.e. process of time passing such that , its effects (time), are identified with Topological changes in space.
- Composition: given and the composite is such that associativity is satisfied :
- Identity: such that and
In this case, given a cobordism the adjoint is simply which is identified with the same manifold as M, but with the direction of "topological changes" reversed.
Instead, the tensor product of and is defined by where is interpreted as the disjoint union of the space manifolds S and .
Subsections
Cecilia Flori 2006-11-19