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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?
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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