Magnitude (Leinster, 2008) is a common categorical generalization of cardinality and of the Euler characteristic of a simplicial complex. It applies to enriched categories, of which metric spaces are a notable example, and in that case gives a new isometric invariant of metric spaces (Leinster, 2013). Applied to infinite metric spaces, this metric invariant—somehow surprisingly—encodes a lot of nontrivial geometric information, such as Minkowski dimension, volume, surface area, etc.(Meckes, 2015; Barceló & Carbery, 2018; Gimperlein & Goffeng, 2021). Partial differential equations, pseudodifferential operators and potential theory have played an important role in establishing these results.

In joint work with Stephanie Chen (Chen & Vigneaux, 2023) (SURF program 2022), we gave a new formula for the magnitude of a finite category $\cat{A}$ in terms of the pseudoinverse of the matrix \begin{equation} \zeta:\Ob\cat{A}\times \Ob \cat{A}\to \Zz, \, (a,b)\mapsto |\Hom(a,b)|. \end{equation} This was closer to the definition for posets (Rota, 1964) that had inspired Leinster. Our work also rederived algebraic properties of the magnitude from properties of the pseudoinverse.

In (Vigneaux, 2024) I propose a novel combinatorial interpretation for the inverse or pseudoinverse of $\zeta$, along the lines of (Brualdi & Cvetkovic, 2008). The interpretation generalizes a celebrated theorem by Philip Hall (Rota, 1964): \begin{equation} \zeta^{-1}(a,b)=\sum_{k\geq 0} (-1)^k \# \{ \text{nondegenerate paths between }a\text{ and }b \} \end{equation} when $a$ and $b$ are elements of a finite poset (in this case $\zeta$ is invertible; its inverse is known as Möbius function).

What does this have to do with information? Following Boltzmann ideas, entropy can be seen as an extension of cardinality: when all elements of a finite set $X$ are equiprobable, the entropy is $\ln |X|$. In turn, magnitude is a generalization of cardinality, and it is natural to introduce a probabilistic extension of it: “categorical entropy”. Stephanie and I (Chen & Vigneaux, 2023) proposed that categorical entropy is defined on finite categories equipped with a probability $p$ on objects and a “kernel” $\theta:\Ob\cat{A} \times \Ob \cat{A} \to [0,\infty)$ such that $\theta(a,a’)=0$ whenever $a\not\to a’$ via the formula \begin{equation}\label{eq:cat_entropy} \mathcal H(A,p,\theta) = - \sum_{a\in \Ob \cat A} p(a) \ln \left(\sum_{b\in \Ob \cat A} \theta(a,b)p(b) \right). \end{equation} This function shares many “nice” properties with Shannon entropy. In the context of metric spaces equipped with probability, \eqref{eq:cat_entropy} appears as a measure of diversity between species when $p$ is its relative abundance and $\theta$ measures their similarity (Leinster & Cobbold, 2012).

Presentations:

Bibliography

  1. Leinster, T. (2008). The Euler Characteristic of a Category. Documenta Mathematica, 13, 21–49.
  2. Leinster, T. (2013). The magnitude of metric spaces. Documenta Mathematica, 18, 857–905.
  3. Meckes, M. W. (2015). Magnitude, diversity, capacities, and dimensions of metric spaces. Potential Analysis, 42, 549–572.
  4. Barceló, J. A., & Carbery, A. (2018). On the magnitudes of compact sets in Euclidean spaces. American Journal of Mathematics, 140(2), 449–494.
  5. Gimperlein, H., & Goffeng, M. (2021). On the magnitude function of domains in Euclidean space. American Journal of Mathematics, 143(3), 939–967.
  6. Chen, S., & Vigneaux, J. P. (2023). A formula for the categorical magnitude in terms of the Moore-Penrose pseudoinverse. Bulletin of the Belgian Mathematical Society - Simon Stevin, 30(3), 341–353.
  7. Rota, G.-C. (1964). On the foundations of combinatorial theory I. Theory of Möbius functions. Probability Theory and Related Fields, 2(4), 340–368.
  8. Vigneaux, J. P. (2024). A combinatorial approach to categorical Möbius inversion and pseudoinversion. ArXiv Preprint 2407.14647.
  9. Brualdi, R. A., & Cvetkovic, D. (2008). A Combinatorial Approach to Matrix Theory and Its Applications. CRC Press. https://books.google.com/books?id=pwx6t8QfZU8C
  10. Leinster, T., & Cobbold, C. A. (2012). Measuring diversity: the importance of species similarity. Ecology, 93(3), 477–489.