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Categorical Magnitude and Entropy

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Geometric Science of Information (GSI 2023)

Part of the book series: Lecture Notes in Computer Science ((LNCS,volume 14071))

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Abstract

Given any finite set equipped with a probability measure, one may compute its Shannon entropy or information content. The entropy becomes the logarithm of the cardinality of the set when the uniform probability is used. Leinster introduced a notion of Euler characteristic for certain finite categories, also known as magnitude, that can be seen as a categorical generalization of cardinality. This paper aims to connect the two ideas by considering the extension of Shannon entropy to finite categories endowed with probability, in such a way that the magnitude is recovered when a certain choice of “uniform” probability is made.

SC acknowledges the support of Marcella Bonsall through her SURF fellowship.

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References

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Author information

Authors and Affiliations

  1. California Institute of Technology, Pasadena, CA, 91125, USA

    Stephanie Chen

  2. Department of Mathematics, California Institute of Technology, Pasadena, CA, 91125, USA

    Juan Pablo Vigneaux

Authors
  1. Stephanie Chen
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  2. Juan Pablo Vigneaux
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Corresponding author

Correspondence to Juan Pablo Vigneaux .

Editor information

Editors and Affiliations

  1. Sony Computer Science Laboratories Inc., Tokyo, Japan

    Frank Nielsen

  2. THALES Land and Air Systems, Meudon, France

    Frédéric Barbaresco

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© 2023 The Author(s), under exclusive license to Springer Nature Switzerland AG

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Chen, S., Vigneaux, J.P. (2023). Categorical Magnitude and Entropy. In: Nielsen, F., Barbaresco, F. (eds) Geometric Science of Information. GSI 2023. Lecture Notes in Computer Science, vol 14071. Springer, Cham. https://doi.org/10.1007/978-3-031-38271-0_28

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  • DOI: https://doi.org/10.1007/978-3-031-38271-0_28

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  • Publisher Name: Springer, Cham

  • Print ISBN: 978-3-031-38270-3

  • Online ISBN: 978-3-031-38271-0

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