Algebras in fusion categories: species and hereditary algebras

We initiate the study of non-semisimple algebras in fusion categories by establishing the framework of $\mathcal{C}$-species – analogous to the framework of species and quivers used in the study of Artin algebras. Under the (necessary) assumption that the fusion category is separable, we show that any algebra is Morita equivalent to an admissible quotient of the path algebra of a $\mathcal{C}$-species. Moreover, we show that an algebra is hereditary if and only if no further quotient is required. These results generalise that of Gabriel’s for finite-dimensional algebras.