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

String diagrams for symmetric powers I: In symmetric monoidal $\mathbb{Q}_{\ge 0}$-linear categories