A hyperoperation on a set M is an operation that associates to each pair of elements a subset of M. Hypergroups and hyperrings are two examples of structures defined in terms of hyperoperations. While they were respectively defined in the 1930s and 1950s, they have recently gained prominence through various appearances in number theory, combinatorics, and absolute algebraic geometry. However, to date there has been relatively little attention given to categories of hyperstructures. I will discuss several categories of hyperstructures that generalize hypergroups. A common theme is that in order for these categories to enjoy good properties like (co)completeness, we must allow for the product or sum of two elements to be the empty subset, which cannot occur in a hypergroup. In particular, I will introduce a category of hyperstructures called mosaics whose subcategory of commutative objects possess a closed monoidal structure reminiscent of the tensor product of abelian groups. This is joint work with So Nakamura.