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 \(\mathcal {C}\) 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 computability models are shown to be special cases of a computability model over a category \(\mathcal {C}\) with a so-called base of computability and a pullback preserving, covariant presheaf on \(\mathcal {C}\). 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.
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Notes
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For a discussion on “strict vs. lax” see [8], Section 2.1.
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If we write \((\texttt {WP} _2)\) as the implication: if \(x \in \mathrm {dom}\big (\langle \langle f, g \rangle \rangle \big )\), then \([\texttt {pr} _{\sigma } \circ \big (\langle \langle f, g \rangle \rangle \big )](x) = f(x)\) and \([\texttt {pr} _{\tau } \circ \big (\langle \langle f, g \rangle \rangle \big )](x) = g(x)\), we need to use the definition of composition of partial functions. As it is noted in [9], p. 53, a computability model with weak products is equivalent to one with standard products.
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If \(F :\mathcal {C} \rightarrow \mathcal {D}\) such that \(T \circ F = S\), where \(S :\mathcal {C} \rightarrow \texttt {Set} \) and \(T :\mathcal {D} \rightarrow \texttt {Set} \), we cannot show, in general, that F preserves monos. It does, if, e.g., T is injective on arrows.
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Acknowledgment
Our research was supported by LMUexcellent, funded by the Federal Ministry of Education and Research (BMBF) and the Free State of Bavaria under the Excellence Strategy of the Federal Government and the Länder.
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Petrakis, I. (2022). Computability Models over Categories and Presheaves. In: Artemov, S., Nerode, A. (eds) Logical Foundations of Computer Science. LFCS 2022. Lecture Notes in Computer Science(), vol 13137. Springer, Cham. https://doi.org/10.1007/978-3-030-93100-1_16
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