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Computability Models over Categories and Presheaves

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Logical Foundations of Computer Science (LFCS 2022)

Part of the book series: Lecture Notes in Computer Science ((LNTCS,volume 13137))

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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

  1. 1.

    This notion is rooted in previous work of Longley in [6,7,8], and is influenced by the work of Cockett and Hofstra in [3] and [4] (see [9], p. 52).

  2. 2.

    In [9] this category is defined for lax computability models, but, as it is remarked in [9], p. 91, the definition makes sense for an arbitrary computability model.

  3. 3.

    For a discussion on “strict vs. lax” see [8], Section 2.1.

  4. 4.

    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.

  5. 5.

    As we don’t define categories of partial arrows, we avoid the equivalence relation between them. A partial function between sets is a partial arrow in \(\texttt {Set} \), also in accordance with the notion of partial function in Bishop set theory (see [10, 11]).

  6. 6.

    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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  1. University of Munich, Munich, Germany

    Iosif Petrakis

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  1. Iosif Petrakis
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Correspondence to Iosif Petrakis .

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Editors and Affiliations

  1. CUNY Graduate Center, New York, NY, USA

    Sergei Artemov

  2. Cornell University, Ithaca, NY, USA

    Anil Nerode

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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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  • DOI: https://doi.org/10.1007/978-3-030-93100-1_16

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