  This work creates two categories of "array-weighted sets" for the purposes of
constructing universal matrix-normed spaces and algebras. These universal
objects have the analogous universal property to the free vector space, lifting
maps completely bounded on a generation set to a completely bounded linear map
of the matrix-normed space.
  Moreover, the universal matrix-normed algebra is used to prove the existence
of a free product for matrix-normed algebras using algebraic methods.
