Levels of selection and multilevel evolutionary processes are essential concepts in evolutionary theory, and yet there is a lack of common mathematical models for these core ideas. Here, we propose a unified mathematical framework for formulating and optimizing multilevel evolutionary processes and genetic algorithms over arbitrarily many levels based on concepts from category theory and population genetics. We formulate a multilevel version of the Wright-Fisher process using this approach, and we show that this model can be analyzed to clarify key features of multilevel selection. Particularly, we derive an extended multilevel probabilistic version of Price's Equation via the Kantorovich Monad, and we use this to characterize regimes of parameter space within which selection acts antagonistically or cooperatively across levels. Finally, we show how our framework can provide a unified setting for learning genetic algorithms (GAs), and we show how we can use a Variational Optimization and a multi-level analogue of coalescent analysis to fit multilevel GAs to simulated data.

翻译：选择层级与多级进化过程是进化理论中的核心概念，然而目前缺乏对这些核心思想的通用数学模型。本文基于范畴论与群体遗传学的概念，提出一个统一的数学框架，用于表述和优化任意多层级上的多级进化过程与遗传算法。我们运用该方法构建了多层级版本的赖特-费希尔过程，并证明该模型可通过分析阐明多级选择的关键特征。特别地，我们通过康托罗维奇单子推导出普莱斯方程的多层级概率扩展形式，并借此刻画了参数空间中不同层级间选择作用呈现拮抗或协同关系的机制。最后，我们展示了该框架如何为遗传算法的学习提供统一环境，并通过变分优化及多层级溯祖分析的模拟方法，实现了多级遗传算法在模拟数据中的拟合。