Categorical representation of DRC-semigroups

DRC-semigroups model associative systems with domain and range operations, and contain many important classes, such as inverse, restriction, Ehresmann, regular ⁎-, and ⁎-regular semigroups; concrete examples include diagram monoids, linear monoids, relation monoids, among many others. In this paper we show that the category of DRC-semigroups is isomorphic to a category of certain biordered categories whose object sets are projection algebras in the sense of Jones. This extends the recent groupoid approach to regular ⁎-semigroups of the first and third authors. We also establish the existence of free DRC-semigroups by constructing a left adjoint to the forgetful functor into the category of projection algebras.