We present two generalizations of the Span construction. The first generalization gives Span of a category with all pullbacks as a (weak) double category. This double category Span A can be viewed as the free double category on the vertical category A where every vertical arrow has both a companion and a conjoint (and these companions and conjoints are adjoint to each other). Thus defined, Span : Cat --> Doub becomes a 2-functor, which is a partial left bi-adjoint to the forgetful functor Vrt : Doub --> Cat, which sends a double category to its category of vertical arrows.
The second generalization gives Span of an arbitrary category as an oplax normal double category. The universal property can again be given in terms of companions and conjoints and the presence of their composites. Moreover, Span A is universal with this property in the sense that Span : Cat --> OplaxNDoub is left bi-adjoint to the forgetful functor which sends an oplax double category to its vertical arrow category.
Keywords: Double categories, Span construction, Localizations, Companions, Conjoints, Adjoints
2000 MSC: 18A40, 18C20, 18D05
Theory and Applications of Categories,
Vol. 24, 2010,
No. 13, pp 302-377.