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The Aldous-Hoover Theorem concerns an infinite matrix of random variables whose distribution is invariant under finite permutations of rows and columns. It states that, up to equality in distribution, each random variable in the matrix can be expressed as a function only depending on four key variables: one common to the entire matrix, one that encodes information about its row, one that encodes information about its column, and a fourth one specific to the matrix entry. We state and prove the theorem within a category-theoretic approach to probability, namely the theory of Markov categories. This makes the proof more transparent and intuitive when compared to measure-theoretic ones. A key role is played by a newly identified categorical property, the Cauchy--Schwarz axiom, which also facilitates a new synthetic de Finetti Theorem. We further provide a variant of our proof using the ordered Markov property and the d-separation criterion, both generalized from Bayesian networks to Markov categories. We expect that this approach will facilitate a systematic development of more complex results in the future, such as categorical approaches to hierarchical exchangeability., Comment: 39 pages

Markov categories have recently emerged as a powerful high-level framework for probability theory and theoretical statistics. Here we study a quantum version of this concept, called involutive Markov categories. First, we show that these are equivalent to Parzygnat's quantum Markov categories but argue that they are simpler to work with. Our main examples of involutive Markov categories involve C*-algebras (of any dimension) as objects and completely positive unital maps as morphisms in the picture of interest. Second, we prove a quantum de Finetti theorem for both the minimal and the maximal C*-tensor norms, and we develop a categorical description of such quantum de Finetti theorems which amounts to a universal property of state spaces., Comment: 49 pages

Markov categories have recently turned out to be a powerful high-level framework for probability and statistics. They accommodate purely categorical definitions of notions like conditional probability and almost sure equality, as well as proofs of fundamental results such as the Hewitt-Savage 0/1 Law, the de Finetti Theorem and the Ergodic Decomposition Theorem. In this work, we develop additional relevant notions from probability theory in the setting of Markov categories. This comprises improved versions of previously introduced definitions of absolute continuity and supports, as well as a detailed study of idempotents and idempotent splitting in Markov categories. Our main result on idempotent splitting is that every idempotent measurable Markov kernel between standard Borel spaces splits through another standard Borel space, and we derive this as an instance of a general categorical criterion for idempotent splitting in Markov categories., Comment: 84 pages (including 18 page appendix and many string diagrams). v2: Corollary 4.4.10 and results needed to establish it were added

We study the positivity and causality axioms for Markov categories as properties of dilations and information flow in Markov categories, and in variations thereof for arbitrary semicartesian monoidal categories. These help us show that being a positive Markov category is merely an additional property of a symmetric monoidal category (rather than extra structure). We also characterize the positivity of representable Markov categories and prove that causality implies positivity, but not conversely. Finally, we note that positivity fails for quasi-Borel spaces and interpret this failure as a privacy property of probabilistic name generation., Comment: 42 pages

Inspired by the intrinsic formality of graded algebras, we prove a necessary and sufficient condition for strongly uniqueness of DG-enhancements. This approach offers a generalization to linearity over any commutative ring. In particular, we obtain several new examples of triangulated categories with a strongly unique DG-enhancement. Moreover, we also show that the bounded derived category of an exact category has a unique enhancement., Comment: 33 pages. I corrected some errors, extended some results (in particular Theorem 7.7) and showed that the bounded derived category of an exact category has a (semi-strongly) unique enhancement (Corollary 3.11 and Proposition 3.12)

We describe how to obtain a global t-structure from a semiorthogonal decomposition with compatible t-structures on every component. This result is used to generalize a well-known theorem of Bondal on full strong exceptional sequences., Comment: 27 pages