### Symmetry, Integrability and Geometry: Methods and Applications (SIGMA)

SIGMA 18 (2022), 075, 27 pages      arXiv:1612.05139      https://doi.org/10.3842/SIGMA.2022.075
Contribution to the Special Issue on Non-Commutative Algebra, Probability and Analysis in Action

### Categorial Independence and Lévy Processes

Malte Gerhold ab, Stephanie Lachs a and Michael Schürmann a
a) Institute of Mathematics and Computer Science, University of Greifswald, Germany
b) Department of Mathematical Sciences, NTNU Trondheim, Norway

Received March 28, 2022, in final form September 30, 2022; Published online October 10, 2022

Abstract
We generalize Franz' independence in tensor categories with inclusions from two morphisms (which represent generalized random variables) to arbitrary ordered families of morphisms. We will see that this only works consistently if the unit object is an initial object, in which case the inclusions can be defined starting from the tensor category alone. The obtained independence for morphisms is called categorial independence. We define categorial Lévy processes on every tensor category with initial unit object and present a construction generalizing the reconstruction of a Lévy process from its convolution semigroup via the Daniell-Kolmogorov theorem. Finally, we discuss examples showing that many known independences from algebra as well as from (noncommutative) probability are special cases of categorial independence.

Key words: general independence; monoidal categories; synthetic probability; noncommutative probability; quantum stochastic processes.

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