Total Maps of Turing Categories

We give a complete characterization of those categories which can arise as the subcategory of total maps of a Turing category. A Turing category provides an abstract categorical setting for studying computability: its (partial) maps may be described, equivalently, as the computable maps of a partial combinatory algebra. The characterization, thus, tells one what categories can be the total functions for partial combinatory algebras. It also provides a particularly easy criterion for determining whether functions, belonging to a given complexity class, can be viewed as the class of total computable functions for some abstract notion of computability.