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When [DELTA] denotes the model of the [DELTA][n] simplexes, [R.sub.[DELTA]] is the identity of [[DELTA].sup.[omicron]]S up to isomorphism. Then what we are doing is to compare at the homotopy level [R.sub.Y] with [1.sub.[[DELTA].sup.[omicron]]S], a notion that allows us to assimilate [R.sub.Y](X) to a "subdivision" of X.
If there is no global isometry carrying g on to g!prime^, then the 'past' does not determine the 'future' even up to isomorphism. Applying Maudlin's metric essentialism, however, we are forced to conclude there is no violation of determinism since at least one of the models portrays the spacetime points represented as having properties contrary to their essences.(8)
Any spacetime model can be characterize up to isomorphism in terms of an algebra starting with an algebraic ring isomorphic to the ring of continuous real-valued functions on the manifold.
It is easy to see that, up to isomorphism, there is exactly one HCS(4) and exactly one HCS(6), both pictured in Figure 2.