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Summary

We show that the concept of entropy is inescapable even in “pure” branches of mathematics such as algebra, topology and category theory. Specifically, we describe an entirely general categorical construction which, when given the real line and the standard simplices as inputs, produces Shannon entropy as the output. The construction involves operads and their algebras. We then show how this categorical line of thinking leads naturally to the entirely concrete and explicit characterization of information loss in Chapter 10.

We add sequential operations to the categorical algebra of weighted andMarkov automata introduced in [L. de Francesco Albasini, N. Sabadini and R.F.C. Walters,  arXiv:0909.4136]. The extra expressiveness of the algebra permits the description of hierarchical systems, and ones withevolving geometry. We make a comparison with the probabilistic automata of Lynch et al. [SIAM J. Comput.37 (2007) 977–1013].

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12 - The Categorical Origins of Entropy

Summary

Appendix A - Karoubian and monoidal categories

The compositional constructionof Markov processes II

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3 results

12 - The Categorical Origins of Entropy

Summary

Appendix A - Karoubian and monoidal categories

The compositional constructionof Markov processes II