  We introduce a notion of "hopfish algebra" structure on an associative
algebra, allowing the structure morphisms (coproduct, counit, antipode) to be
bimodules rather than algebra homomorphisms. We prove that quasi-Hopf algebras
are examples of hopfish algebras. We find that a hopfish structure on the
commutative algebra of functions on a finite set G is closely related to a
"hypergroupoid" structure on G. The Morita theory of hopfish algebras is also
discussed.
