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Entropy is inevitable

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The title refers not to the death of the universe, but to the fact that the concept of entropy is present in the pure-mathematical heartlands of algebra and topology, whether we like it or not. I will describe a categorical machine which, when fed as input the concepts of topological simplex and real number, produces as output the concept of Shannon entropy. The most important component of this machine is the notion of “internal algebra” in an algebra for an operad (generalizing the notion of monoid in a monoidal category). There is more: the resulting characterization of Shannon entropy can be stripped completely of its categorical garb, giving a simple, new, and entirely elementary characterization. This last theorem is joint with John Baez and Tobias Fritz.

This talk is part of the Category Theory Seminar series.

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