In the 1970s, Gabriel (and later Bernstein—Gelfand—Ponomarev) discovered a deep connection between quiver representations and root systems associated to Weyl groups of (simply-laced) Lie algebras. In particular, indecomposable quiver representations and positive roots are in one-to-one correspondence, with the ADE Dynkin diagrams appearing in both worlds. On the other hand, Weyl groups are a family of Coxeter groups, which themselves come with a finite-type classification: a Coxeter group is finite if and only if its corresponding Coxeter graph is a Coxeter—Dynkin diagram. In this talk I will introduce a generalisation of quivers whose representations are related to root systems of Coxeter groups. As we will see, fusion categories, a special kind of monoidal categories (rigid, finite, semisimple), will be the key player. If time allows, I will speak about how algebras (monoids) in fusion categories are related to this story — similar to the relationship between finite dimensional algebras (monoids in the category of vector spaces) and quivers. This talk will be based on joint work with Ben Elias arXiv:2404.09714.