Strict computability models over categories and presheaves

Abstract

Abstract Generalizing slightly the notions of a strict computability model and of a simulation between them, which were elaborated by Longley and Normann (2015, Higher-Order Computability), we define canonical strict computability models over certain categories and appropriate presheaves on them. We study the canonical total computability model over a category $\mathcal {C}$ and a covariant presheaf on $\mathcal {C}$ and the canonical partial computability model over a category $\mathcal {C}$ with pullbacks and a pullback preserving, covariant presheaf on $\mathcal {C}$. These strict computability models are shown to be special cases of a strict computability model over a category $\mathcal {C}$ with a so-called base of computability and a pullback preserving, covariant presheaf on $\mathcal {C}$, connecting in this way Rosolini’s theory of dominions with the theory of computability models. All our notions and results are dualized by considering certain (contravariant) presheaves on appropriate categories.