  In order to diagnose the cause of some defects in the category of canonical
hypergroups, we investigate several categories of hyperstructures that
generalize hypergroups. By allowing hyperoperations with possibly empty
products, one obtains categories with desirable features such as completeness
and cocompleteness, free functors, regularity, and closed monoidal structures.
We show by counterexamples that such constructions cannot be carried out within
the category of canonical hypergroups. This suggests that (commutative) unital,
reversible hypermagmas -- which we call mosaics -- form a worthwhile
generalization of (canonical) hypergroups from the categorical perspective.
Notably, mosaics contain pointed simple matroids as a subcategory, and
projective geometries as a full subcategory.

Category Theory;Combinatorics;Group Theory;Rings and Algebras