Bart Jacobs, Sam Staton
This paper reformulates a classical result in probability theory from the 1930s in modern categorical terms: de Finetti's representation theorem is redescribed as limit statement for a chain of finite spaces in the Kleisli category of the Giry monad. This new limit is used to identify among exchangeable coalgebras the final one.
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We present a novel proof of de Finetti's Theorem characterizing permutation-invariant probability measures of infinite sequences of variables, so-called exchangeable measures. The proof is phrased in the language of Markov categories, which provide an abstract categorical framework for probability and information flow. The diagrammatic and abstract nature of the arguments makes the proof intuitive and easy to follow. We also show how the usual measure-theoretic version of de Finetti's Theorem for standard Borel spaces is an instance of this result.
Absolute continuity, supports and idempotent splitting in categorical probability
在马尔可夫范畴中发展了绝对连续性和支撑的概念，并研究了幂等分裂，证明了标准Borel空间上的幂等可测马尔可夫核可分裂为另一标准Borel空间。
Empirical Measures and Strong Laws of Large Numbers in Categorical Probability
本文提出两个公理研究经验测度的极限，在范畴概率中证明了de Finetti定理、Glivenko-Cantelli定理和强大数定律的抽象版本，并通过具体构造恢复了标准定理。
The Aldous$\unicode{x2013}$Hoover Theorem in Categorical Probability
本文从Markov范畴理论出发，证明了Aldous-Hoover定理，并引入了Cauchy-Schwarz公理，同时探讨了其在分层可交换性中的应用。
Categorical probability spaces, ergodic decompositions, and transitions to equilibrium
该研究构建了概率空间与保测马尔可夫核的范畴框架，证明几乎必然不变σ-代数的极限/余极限性质，给出遍历分解定理的范畴版本，并将多个经典概率定理纳入统一的极限-余极限图景。
We present a categorical viewpoint of probability measures by showing that a probability measure can be viewed as a weakly averaging affine measurable functional taking values in the unit interval which preserves limits. The probability measures on a space are the elements of a submonad of a double dualization monad on the category of measurable spaces into the unit interval, and this monad is naturally isomorphic to the Giry monad. We show this submonad is the codensity monad of a functor from the category of convex spaces to the category of measurable spaces. A theorem proving the integral o…
Enhanced 2-categorical structures, two-dimensional limit sketches and the symmetry of internalisation
该研究从增强型2-范畴视角解释二维范畴论中结构的内在严格性，引入增强型2-范畴极限草图以弥补现有理论不足，并建立了这类结构的内化对称性。
A categorical treatment of the Radon-Nikodym theorem and martingales
本文用范畴论方法证明Radon-Nikodym定理，通过自然同构与Kan延拓推广至一般概率空间，并利用右Kan延拓中的条件期望范畴化表示鞅，进而在丰富范畴中证明鞅收敛定理。
In this paper I develop categorical foundations needed for a rigorous approach to the definition of conformal field theory outlined by Graeme Segal. I discuss pseudo algebras over theories and 2-theories, their pseudo morphisms, bilimits, bicolimits, biadjoints, stacks, and related concepts.
In this paper we prove that various quasi-categories whose objects are $\infty$-categories in a very general sense are complete: admitting limits indexed by all simplicial sets. This result and others of a similar flavor follow from a general theorem in which we characterize the data that is required to define a limit cone in a quasi-category constructed as a homotopy coherent nerve. Since all quasi-categories arise this way up to equivalence, this analysis covers the general case. Namely, we show that quasi-categorical limit cones may be modeled at the point-set level by pseudo homotopy limit…
Fraïssé limit and Ramsey Theorem: the case of MV-algebras and a categorical generalization
该研究刻画了有限MV-代数的Fraïssé极限并证明其满足Ramsey性质，还将结论推广至一般范畴，得出满足Kechris-Pestov-Todorcevic对应最低条件时，有限结构范畴与其完备子范畴的Ramsey性质等价的结论。
Categorical (Co)Limits of Quantum Graphs
本文通过量子图作为左理想的新表征，提出不依赖表示的量子图映射概念，并引入量子图的范畴（余）极限，同时探讨图的另一种量子化方式。
Categorical approach to graph limits
本文提出图极限的范畴框架，通过定义对象和态射，结合范畴论工具证明图极限空间的紧性，并引入基于s-收敛的收敛概念。
It is possible to translate a modified version of K. Worytkiewicz's combinatorial semantics of CCS (Milner's Calculus of Communicating Systems) in terms of labelled precubical sets into a categorical semantics of CCS in terms of labelled flows using a geometric realization functor. It turns out that a satisfactory semantics in terms of flows requires to work directly in their homotopy category since such a semantics requires non-canonical choices for constructing cofibrant replacements, homotopy limits and homotopy colimits. No geometric information is lost since two precubical sets are isomor…
Given a limit sketch in which the cones have a finite connected base, we show that a model structure of "up to homotopy" models for this limit sketch in a suitable model category can be transferred to a Quillen equivalent model structure on the category of strict models. As a corollary of our general result, we obtain a rigidification theorem which asserts in particular that any $Θ_n$-space in the sense of Rezk is levelwise equivalent to one that satisfies the Segal conditions on the nose. There are similar results for dendroidal spaces and $n$-fold Segal spaces.
The categorical basis of dynamical entropy
该研究从范畴论视角研究拓扑动力熵，提出合格函子对工具证明紧空间开覆盖的直径与Lebesgue数构成该类函子对，揭示多种迭代复杂度概念收敛到共同拓扑熵是结构假设的结果。
Recently, the author and Bob Coecke have introduced a categorical formulation of Quantum Mechanics. In the present paper, we shall use it to open up a novel perspective on No-Cloning. What we shall find, quite unexpectedly, is a link to some fundamental issues in logic, computation, and the foundations of mathematics. A striking feature of our results is that they are visibly in the same genre as a well-known result by Joyal in categorical logic showing that a `Boolean cartesian closed category' trivializes, which provides a major road-block to the computational interpretation of classical log…
Towards concept analysis in categories: limit inferior as algebra, limit superior as coalgebra
本文用范畴论中的上极限和下极限运算刻画普通范畴的 Dedekind-MacNeille 完备化，统一了富集矩阵双完备化与普通范畴完备化之间的差距，为范畴化概念挖掘与分析开辟了新路径。
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