### New paper with Steve Lack and Richard Wood - bicategories of spans

We have just put a new paper on arxiv called "Bicategories of spans as cartesian bicategories", where we characterize such bicategories as follows: a bicategory is biequivalent to Span(E) for E a category with finite limits iff it is cartesian, each comonad has an Eilenberg-Moore object, and every map is comonadic.

There is quite a bit of further stuff in the paper, but one extra point I would like to mention as it is relevant to our paper on spans and cospans.

The extra result is that Span(E) for E finitely complete has direct sums iff E is extensive. This is related to work of Heindel and Sobocinski (T. Heindel and P. Sobocinski. Van Kampen colimits as bicolimits in Span. In Calco '09, LNCS, Springer, 2009).

To bring this result down to earth lets look at Span(Sets) and see why it has direct sums. Consider a span R from X to Y. Suppose X=U1+U2+..Um, and Y=V1+...Vn. Then clearly R breaks up into a family of sets Rij, the elements which go from Ui to Vj. So the span R becomes a matrix Rij of spans. A special case of this is when R is an endomorphism of X, that is, R is a graph. If the vertices of the graph break up as a sum U1+...Um, the graph may be expressed as a square matrix of spans Ui->Uj (i,j=1,2,...,m). (A span in sets is a kind of bipartite graph.)

There is quite a bit of further stuff in the paper, but one extra point I would like to mention as it is relevant to our paper on spans and cospans.

The extra result is that Span(E) for E finitely complete has direct sums iff E is extensive. This is related to work of Heindel and Sobocinski (T. Heindel and P. Sobocinski. Van Kampen colimits as bicolimits in Span. In Calco '09, LNCS, Springer, 2009).

To bring this result down to earth lets look at Span(Sets) and see why it has direct sums. Consider a span R from X to Y. Suppose X=U1+U2+..Um, and Y=V1+...Vn. Then clearly R breaks up into a family of sets Rij, the elements which go from Ui to Vj. So the span R becomes a matrix Rij of spans. A special case of this is when R is an endomorphism of X, that is, R is a graph. If the vertices of the graph break up as a sum U1+...Um, the graph may be expressed as a square matrix of spans Ui->Uj (i,j=1,2,...,m). (A span in sets is a kind of bipartite graph.)

Labels: category theory, mathematics, Optimistic

## 2 Comments:

That's a pretty result, Bob. I plan on mentioning this in the nLab some time soon.

Very interesting paper.

But a question:

Is this the paper

"[CKWW] A. Carboni, G.M. Kelly, R.F.C. Walters and R.J. Wood. Cartesian bicategories III. To appear."

that was mentioned in the references to

"Cartesian Bicategories II", TAC Vol. 19 (2008)?

(Considering that Carboni and Kelly are now deceased, and the subject of this paper, that seems a reasonable question.)

If not, what is the status of CB III?

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