Recall the notion of a unitary pre-tabular allegory. Bicategories of relations are equivalent, but this has yet to be shown rigorously and published, as far as I’m aware.
This PDF file gives an explicit argument, which I put at bicategory of relations until Mike Shulman suggested the neater proof that’s there now. But I’ll keep this here anyway.
The following lemma is used at allegory.
If and are maps in a unitary pre-tabular allegory, then
Let . Then and , by the modular law and the fact that projections tabulate top elements. So if and only if it is a map.
For the unit inequality, we have
The second equality follows from distributivity , which is an equality because .
Revised on November 6, 2012 04:34:14
by Finn Lawler?