Free quantaloids

AbstractQuantaloids are categories enriched in the symmetric, monoidal closed category of sup-lattices and they are a natural generalization of quantales. This article looks at the free quantaloid P(A) generated by a locally small category A and how it arises from a monad on the category of locally small categories. The notion of quantic nucleus on a quantale is generalized to quantaloids and finally we look at categories enriched in P(A) and P(A)-functors and bimodules and their relationship with lax functors Aop→R, where R is the category of sets and relations