Abstract
We define the notion of a measure family: a pre-cosheaf of finite measures over a finite set; every joint measure on a product of finite sets has an associated measure family. To each measure family there is an associated index, or “Euler characteristic”, related to the Tsallis deformation of mutual information. This index is further categorified by a (weighted) simplicial complex whose topology retains information about the correlations between various subsystems.
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Notes
- 1.
See [15] for a precise categorical equivalence between commutative W\(^*\)-algebras and (localizable) measurable spaces.
- 2.
The empty measure corresponds to the zero expectation value on the zero algebra.
- 3.
If one works with probability measures and measure-preserving maps, \(\boxplus \) instead manifests as an operadic structure which encapsulates the ability to take convex linear combinations of probability measures; this is the approach taken by [6].
- 4.
It is easy to write down a natural bijection of \([\textsf{Meas}]\) with \(\coprod _{n=0}^{\infty } (\mathbb {R}_{\ge 0})^{\times n}\), taking \(\mathbb {R}^{\times 0} {:}{=}\{\star \}\) to correspond to the empty measure. This observation can be used to equip \([\textsf{Meas}]\) with a topology as in [3].
- 5.
In some sense a 2-measure is an “acyclic cosheaf” of measures.
- 6.
One can define an index with respect to any cover of \(P_{\mu }\); but our primary interest will be the cover that is the complement of the finest partition of \(P_{\upmu }\).
- 7.
Augmented in this context means there is an additional degree \(-1\) component and a single map from the degree 0 component to the degree \(-1\) component.
- 8.
All interesting quantities are equivariant under change of total order.
- 9.
If \(\mu _{P}\) does not vanish everywhere, then \(| {\texttt{S}} \mu _{\emptyset }| = |\{\star \}| = 1\). Consequently, one can show that \(\mathfrak {X}_{0}[\texttt{A}^{\boldsymbol{\mu }}] = 1 - \chi (\varDelta '_{\upmu })\).
- 10.
This is a specialization of a functor from (localizable) measurable spaces to the Banach space underlying the W\(^*\)-algebra of essentially bounded measurable functions.
- 11.
The classical analog of the GNS and the commutant complexes of [11] both reduce to the alternating sum of face maps complex that is used in this note.
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Mainiero, T. (2023). Higher Information from Families of Measures. 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_25
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