Symmetric powers are an important notion in mathematics, computer science and physics. In mathematics, they are used to build symmetric algebras, in computer science, to build free exponential modalities of linear logic and in physics, Fock spaces. We study symmetric powers through the lens of category theory. We focus here on the simpler case where nonnegative rational scalars are available ie. we study symmetric powers in symmetric monoidal $\mathbb{Q}_{\ge 0}$-linear categories. Among the developments, a main point is the introduction of the notion of binomial graded bimonoid and the associated string diagrams which characterize symmetric powers in this setting.

翻译：对称幂是数学、计算机科学和物理学中的一个重要概念。在数学中，它用于构建对称代数；在计算机科学中，用于构建线性逻辑的自由指数模态；在物理学中，则用于构建福克空间。我们通过范畴论的视角研究对称幂，并聚焦于非负有理标量可用的简单情形，即在对称幺半$\mathbb{Q}_{\ge 0}$-线性范畴中研究对称幂。主要进展之一在于引入了二项式分次双幺半群的概念及其相关的弦图，这些弦图在此框架下刻画了对称幂的特征。