Universal properties of impure programming languages

We investigate impure, call-by-value programming languages. Our first language only has variables and let-binding. Its equational theory is a variant of Lambek's theory of multicategories that omits the commutativity axiom. We demonstrate that type constructions for impure languages --- products, sums and functions --- can be characterized by universal properties in the setting of 'premulticategories', multicategories where the commutativity law may fail. This leads us to new, universal characterizations of two earlier equational theories of impure programming languages: the premonoidal categories of Power and Robinson, and the monad-based models of Moggi. Our analysis thus puts these earlier abstract ideas on a canonical foundation, bringing them to a new, syntactic level.