As an example of the categorical apparatus of pseudo algebras over2-theories, we show that pseudo algebras over the 2-theory ofcategories can be viewed as pseudo double categories with folding oras appropriate 2-functors into bicategories. Foldings are equivalentto connection pairs, and also to thin structures if the vertical andhorizontal morphisms coincide. In a sense, the squares of a doublecategory with folding are determined in a functorial way by the2-cells of the horizontal 2-category. As a special case, strict2-algebras with one object and everything invertible are crossedmodules under a group.
Journal of Homotopy and Related Structures, Vol. 2(2007), No. 2, pp. 119-170