UNIVERSAL PROPERTIES OF SPAN Dedicated to Aurelio Carboni on the occasion of his 60 th birthday

We give two related universal properties of the span construction. The first involves sinister morphisms out of the base category and sinister transformations. The second involves oplax morphisms out of the bicategory of spans having an extra property; we call these “jointed” oplax morphisms. Introduction Even before they were formally introduced in [Ka], the importance of adjoint functors was recognized in many individual cases, e.g., free groups, fraction fields, Stone-Čech compactifications, and adjunctions on linear spaces. Any functor that has an adjoint has many important properties; for instance, a functor with a right adjoint preserves colimits [M2]. Of course, most functors (and even many important ones) have neither left nor right adjoints. This motivates the introduction of profunctors [Bé2]. These are generalizations of functors and in the bicategory Prof of categories with profunctors every functor (viewed as a profunctor) has a right adjoint. The reader will note that these adjunctions exist in a bicategory different from the 2-category Cat of categories in which adjunctions were originally defined. Indeed, the usual characterization (or definition) of adjunction in terms of unit and counit satisfying the triangle equalities is a 2-categorical concept, and adding the appropriate isomorphisms makes it into a bicategorical one. The idea of an adjunction is fruitful enough to motivate adding a 2-cell structure to an ordinary category, just so that one can consider adjunctions. This has been done in several ways. Historically, the first instance of this was probably the generalization of the concept of “function” to that of “relation” (before categories were even invented). The idea of relations being ordered by inclusion (i.e., as subsets of the product) also goes back long before the invention of categories, but can be considered as providing a 2-cell structure for the category Rel of sets and relations. In this bicategory, every function determines a relation and as such has a right adjoint, namely the reverse relation. Furthermore, every relation is the composite of one coming from a function with the reverse of one of these, All three authors are suppported by NSERC grants. Received by the editors 2004-04-21 and, in revised form, 2004-05-17. Published on 2004-12-05. 2000 Mathematics Subject Classification: 18A40, 18D05.