## Bicategories of spans as cartesian bicategories

### Richard Wood – 6 February 2008

joint work with Steve Lack and Bob Walters
It is clear that Span(E), for E a category with finite limits, is a cartesian bicategory. In this talk I will go as far as time allows to exhibit further axioms on a cartesian B, valid in Span(E), so as to prove B ~ Span(E) , for some E with finite limits. Since this would entail Map(B) ~ Map(Span(E)) and the latter is well known [Carboni, Kasangian, Street, Walters, ???] to be E, the first task at least is to show that Map(B) is a m e r e category with pullbacks. If every object of B is Frobenius then, as shown last time, all the Map(B)(T,A) are groupoids. We begin this time by showing that if every object of B is s e p a r a b l e then all the Map(B)(T,A) are ordered sets.

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