AI 中文总结
研究引入具有凸双积的范畴，指出其与有限双积范畴在矩阵演算上的差异，产生适用于概率设置的框架，利用此联系建立同构支撑概率带图，并对公理概率布尔电路有效。
AI 中文摘要
具有有限双积的范畴在范畴论中起着核心作用，为统一研究加法和线性结构提供了抽象框架。本文引入具有“凸”双积的范畴，直观上是将线性结构限制为凸结构。我们表明，具有有限双积的范畴产生基于任意线性组合的矩阵演算，而凸双积范畴则诱导基于随机（更一般地，次随机）矩阵的矩阵演算。这一观点产生了一个针对概率设置的精细代数和组合框架。我们利用这种联系建立了一个同构，它支撑概率带图，一种用于双幺半（也称为rig）范畴的图形形式，并通过提供概率布尔电路的完整公理来证明其有效性。
英文摘要
Categories with finite biproducts play a central role in category theory, providing an abstract setting in which additive and linear structures can be studied uniformly. In this paper, we introduce categories with \emph{convex} biproducts, which intuitively restrict the linear structures to convex ones. We show that, whereas categories with finite biproducts give rise to a matrix calculus based on arbitrary linear combinations, convex biproduct categories instead induce a matrix calculus based on stochastic (more generally, substochastic) matrices. This perspective yields a refined algebraic and compositional framework tailored to probabilistic settings. We exploit this connection to establish an isomorphism that underpins probabilistic tape diagrams, a graphical formalism for bimonoidal (also known as rig) categories, and we demonstrate its effectiveness by providing a complete axiomatisation of probabilistic Boolean circuits.
