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.

http://www.tac.mta.ca/tac/volumes/24/13/24-13.dvi

http://www.tac.mta.ca/tac/volumes/24/13/24-13.ps

http://www.tac.mta.ca/tac/volumes/24/13/24-13.pdf

ftp://ftp.tac.mta.ca/pub/tac/html/volumes/24/13/24-13.dvi

ftp://ftp.tac.mta.ca/pub/tac/html/volumes/24/13/24-13.ps