Double categories as double partial monoids, and their involutive structure

A double category has an equivalent description as a double partial monoid, that is, a set equipped with two partial monoid structures such that all structure maps are partial monoid homomorphisms. Double categories have an interesting and natural involutive structure, which we define and characterise in terms of "internal adjointability" functors (or partial monoid homomorphisms). There is a well-known result by Brown, Mosa and Spencer, demonstrating the equivalence of categories between a certain category of double categories and the category of all 2-categories. (A 2-category is equivalent to a double partial monoid plus an extra condition.) With the intrinsic involutive structures taken into account, the situation is more complex and we arrive at a new result involving an isomorphism of categories, (a stronger equivalence). (For people familiar with connections and/or conjoints, conjoint data is equivalent to the connection data plus the horizontal and vertical 2-arrow involutions, for instance: $\Gamma^{*^2_v}$ and $\Gamma^{*^2_h}$.) Involutive double categories are good generalisations of symmetric monoidal categories with duals. For people unfamiliar with categories, we introduce them from the point of view of a monoid with a partially defined multiplication (referring to the book, "Categories for the working mathematician" by Mac Lane). (Joint work with Paolo Bertozzini, and thanks to Roberto Conti and Pedro Resende.)