Introduced in the 1990s in the context of the algebraic approach to graph rewriting, gs-monoidal categories are symmetric monoidal categories where each object is equipped with the structure of a commutative comonoid. They arise for example as Kleisli categories of commutative monads on cartesian categories, and as such they provide a general framework for effectful computation. Recently proposed in the context of categorical probability, Markov categories are gs-monoidal categories where the monoidal unit is also terminal, and they arise for example as Kleisli categories of commutative affine monads, where affine means that the monad preserves the monoidal unit. The aim of this paper is to study a new condition on the gs-monoidal structure, resulting in the concept of weakly Markov categories, which is intermediate between gs-monoidal categories and Markov ones. In a weakly Markov category, the morphisms to the monoidal unit are not necessarily unique, but form a group. As we show, these categories exhibit a rich theory of conditional independence for morphisms, generalising the known theory for Markov categories. We also introduce the corresponding notion for commutative monads, which we call weakly affine, and for which we give two equivalent characterisations. The paper argues that these monads are relevant to the study of categorical probability. A case at hand is the monad of finite non-zero measures, which is weakly affine but not affine. Such structures allow to investigate probability without normalisation within an elegant categorical framework.

翻译：在20世纪90年代针对图重写的代数方法背景下引入的gs-幺半范畴，是每个对象都配备了交换余半群结构的对称幺半范畴。这类范畴例如作为笛卡尔范畴上交换单子的Kleisli范畴出现，从而为有效计算提供了通用框架。近来在范畴概率论背景下提出的马尔可夫范畴，是幺半单位也是终对象的gs-幺半范畴，它们例如作为交换仿射单子的Kleisli范畴出现，其中仿射意指单子保持幺半单位。本文旨在研究gs-幺半结构上的新条件，由此得到弱马尔可夫范畴的概念，该概念介于gs-幺半范畴与马尔可夫范畴之间。在弱马尔可夫范畴中，到幺半单位的态射未必唯一，但构成一个群。如我们所示，此类范畴为态射的条件独立性提供了丰富理论，推广了已知的马尔可夫范畴理论。我们还引入了交换单子的相应概念，称为弱仿射单子，并给出了该概念的两种等价刻画。本文论证这些单子与范畴概率论的研究相关。一个典型实例是有限非零测度单子，它是弱仿射的而非仿射的。此类结构使得我们能够在优雅的范畴论框架下研究未归一化的概率。