Quantum categories were introduced by Day and Street as generalizations of both bi(co)algebroids and small categories. We clarify details of that work. In particular, we show explicitly how the monadic definition of a quantum category unpacks to a set of axioms close to the definitions of a bialgebroid in the Hopf algebraic literature. We introduce notions of functor and natural transformation for quantum categories and consider various constructions on quantum structures.
Keywords: quantum category, monoidal category, comonad
2000 MSC: 18D35
Theory and Applications of Categories, Vol. 25, 2011, No. 1, pp 1-37.