Computability Models over Categories and Presheaves

Abstract

Generalising slightly the notions of a strict computability model and of a simulation between them, which were elaborated by Longley and Normann in [9], we define canonical computability models over certain categories and appropriate presheaves on them. We study the canonical total computability model over a category C and a covariant presheaf on C, and the canonical partial computability model over a category C with pullbacks and a pullback preserving, covariant presheaf on C. These computability models are shown to be special cases of a computability model over a category C with a so-called base of computability and a pullback preserving, covariant presheaf on C. In this way Rosolini's theory of dominions is connected with the theory of computability models. All our notions and results are dualised by considering certain (contravariant) presheaves on appropriate categories.