I have been thinking about Spekkens Toy model in terms of interfaces. The Spekkens paper concerns a physics based on only being able to receive answers to half the number of questions necessary to specify the state of a system. This is something like having a limited interface to some kind of system. I take an apparatus as an internal category in a monoidal category and the apparatus is seen as some limited interface to an underlying quantum causal structure. Would it be possible to reformulate Spekkens' idea in terms of internal categories?
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The short answer is yes: these ideas can be formulated as internal algebras in a monoidal category. Take a look at http://arxiv.org/abs/1003.5005 for starters. Bill Edwards' PhD thesis has quite a bit more, and other papers by him and Bob Coecke may also be of interest.
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$\begingroup$ Thanks Joe. I've been lurking since (before!) it started :-) $\endgroup$– Ross DuncanDec 11, 2011 at 19:03