Adjoining adjoints (Q1404973)

If \(\mathcal{C}\) is a category with pullbacks, the bicategory, \(\text{Span}(\mathcal{C})\), has the universal property that it is the result of freely adjoining right adjoints to the arrows in \(\mathcal{C}\), subject just to the Beck condition. Here the authors describe a 2-category \(\Pi_2\mathcal{C}\), which is similarly the result of freely adjoining right adjoints to the arrows in \(\mathcal{C}\) for any category, \(\mathcal{C}\). The construction uses zig-zag diagrams reminiscent of the Dwyer-Kan hammock localisation from homotopical algebra. The resulting 2-cell diagrams, here called `fences', are very richly structured objects. The 2-cells in \(\Pi_2\mathcal{C}\) are equivalence classes of fences. Of particular interest is \(\Pi_2\mathcal{C}\), when \(\mathcal{C}\) is a free category on a directed graph. In this case the authors explore a beautiful link with planar \((p,q)\)-diagrams (Kauffman diagrams) and thus the tantalising possibility of links with areas of \(C^*\)-algebra theory.