Related papers: Categorial Independence and L\'evy Processes
Fractional Brownian motion can be represented as an integral of a deterministic kernel w.r.t. an ordinary Brownian motion either on infinite or compact interval. In previous literature fractional L\'evy processes are defined by integrating…
We propose a data-driven approach to identifying the functionally independent invariants that can be constructed from a tensor with a given symmetry structure. Our algorithm proceeds by first enumerating graphs, or tensor networks, that…
Free independence is an important tool for studying the structure of operator algebras. It is natural to ask from the model-theoretic standpoint whether free independence is captured well in first-order model theory via the notion of a…
We present an abstract, categorical formulation of dependent functions in a fundamental manner and independently from the Sigma-construction. For that, we define first the notion of a category with family-arrows, or a $\f$-category. A $(\f,…
A symmetric monoidal category naturally arises as the mathematical structure that organizes physical systems, processes, and composition thereof, both sequentially and in parallel. This structure admits a purely graphical calculus. This…
To an exact endofunctor of a triangulated category with a split-generator, the notion of entropy is given by Dimitrov-Haiden-Katzarkov-Kontsevich, which is a (possibly negative infinite) real-valued function of a real variable. In this…
We give a category-theoretic construction of simple and NSOP$_1$-like independence relations in locally finitely presentable categories, and in the more general locally finitely multipresentable categories. We do so by identifying…
In this paper, we examine how various notions of independence in non-commutative probability theory arise in bi-free probability. We exhibit how Boolean and monotone independence occur from bi-free pairs of faces and establish a Kac/Loeve…
Graph independence (also known as $\epsilon$-independence or $\lambda$-independence) is a mixture of classical independence and free independence corresponding to graph products or groups and operator algebras. Using conjugation by certain…
We develop Markov categories as a framework for synthetic probability and statistics, following work of Golubtsov as well as Cho and Jacobs. This means that we treat the following concepts in purely abstract categorical terms: conditioning…
This paper shows how internal models for polymorphic lambda calculi arise in any 2-category with a notion of discreteness. We generalise to a 2-categorical setting the famous theorem of Peter Freyd saying that there are no sufficiently…
Category theory is famous for its innovative way of thinking of concepts by their descriptions, in particular by establishing universal properties. Concepts that can be characterized in a universal way receive a certain quality seal, which…
We introduce a notion of complexity of diagrams (and in particular of objects and morphisms) in an arbitrary category, as well as a notion of complexity of functors between categories equipped with complexity functions. We discuss several…
The randomization of a complete first order theory $T$ is the complete continuous theory $T^R$ with two sorts, a sort for random elements of models of $T$, and a sort for events in an underlying probability space. We study various notions…
We construct a complex entire function with arbitrary number of variables which has the following property: The infinite set consisting of all the values of all its partial derivatives of any orders at all algebraic points, including zero…
Conditional independence and graphical models are well studied for probability distributions on product spaces. We propose a new notion of conditional independence for any measure $\Lambda$ on the punctured Euclidean space $\mathbb…
To formalize calculations in linear algebra for the development of efficient algorithms and a framework suitable for functional programming languages and faster parallelized computations, we adopt an approach that treats elements of linear…
Categorical probability has recently seen significant advances through the formalism of Markov categories, within which several classical theorems have been proven in entirely abstract categorical terms. Closely related to Markov categories…
In this paper we provide a detailed construction of an equivalence between the category of Lawvere theories and the category of relative monads on the obvious functor $Jf:F\rightarrow Sets$ where $F$ is the category with the set of objects…
The deformation cohomology of a tensor category controls deformations of its monoidal structure. Here we describe the deformation cohomology of tensor categories generated by one object (the so-called Schur-Weyl categories). Using this…