Category nCob in General Relativity


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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 $ S_1$ , then, M is a cobordism between S and $ S_1$ iff the boundary of M is the union of S and $ S_1$ . 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.
Within this framework we identify the following:
  1. Composition: given $ M:S\rightarrow S_1$ and $ M^1:S_1\rightarrow S_2$ the composite is $ M^1M:S\rightarrow S_2$ such that associativity is satisfied : $ (M^2M^1)M=M^2(M^1M)$
  2. Identity: $ 1_S:S\rightarrow S$ such that $ 1_S o M=M$ and $ M o 1_S=M$
Similarly as for category Hilb, it can be shown that also nCob is both a *-Category and a Monoidal Category (see [1] for details).
In this case, given a cobordism $ M:S\rightarrow S_1$, the adjoint is simply $ M^*: S_1\rightarrow S$, which is identified with the same manifold as M, but with the direction of "topological changes" reversed.
Instead, the tensor product $ M\otimes M_1$ of $ M:S\rightarrow S_1$ and $ M_1: S_2\rightarrow S_3$ is defined by $ M\otimes M_1: S\otimes S_2\rightarrow S_1\otimes S_2$ where $ S\otimes S_2$ is interpreted as the disjoint union of the space manifolds S and $ S_2$ .
Subsections
Cecilia Flori 2006-11-19