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\begin{document}

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\section*{Turing Category}

\hypertarget{context}{}\subsubsection*{{Context}}\label{context}

\hypertarget{constructivism_realizability_computability}{}\paragraph*{{Constructivism, Realizability, Computability}}\label{constructivism_realizability_computability}

[[!include constructivism - contents]]

\hypertarget{contents}{}\section*{{Contents}}\label{contents}

\noindent\hyperlink{definition}{Definition}\dotfill \pageref*{definition} \linebreak
\noindent\hyperlink{basic_properties}{Basic properties}\dotfill \pageref*{basic_properties} \linebreak
\noindent\hyperlink{more_examples}{More Examples}\dotfill \pageref*{more_examples} \linebreak
\noindent\hyperlink{weak_limits_and_exact_completions}{Weak Limits and Exact Completions}\dotfill \pageref*{weak_limits_and_exact_completions} \linebreak
\noindent\hyperlink{references}{References}\dotfill \pageref*{references} \linebreak


\hypertarget{definition}{}\subsection*{{Definition}}\label{definition}

A Turing category is a certain [[categorification]] of [[partial combinatory algebras]] based on [[restriction categories]].

\begin{defn}
\label{TuringCategory}\hypertarget{TuringCategory}{}
\textbf{(Turing category)}

A \emph{Turing category} is a cartesian restriction category $(\mathcal{C}, \bar{} )$ with a fixed object A and morphism $\bullet : A \times A \rightarrow A$ having the following universal property: for each $\mathcal{C}$-morphism $f : X \rightarrow Y$ there is a section $s: Y \rightarrow A$ and retract $r : A \rightarrow X$, along with a total map $h : \mathbf{1} \rightarrow A$ satisfying the diagram:

\begin{verbatim}\begin{center} 
\begin{tikzcd} \arrow[r, "\bullet"] A \times A & A \arrow[r, "r"] & X \arrow[ddll, "f"] \\
\arrow[u, "\mathsf{id_A} \times h"] A \times 1 \simeq A \arrow[ur, swap,  "sfr"] & & \\
Y \arrow[u, "s"] & & 
\end{tikzcd}
\end{center}\end{verbatim}
\end{defn}
While restriction structure captures the function partiality one finds in [[computability theory]], the universal property stated above captures the universal coding of $\mathcal{C}$-morphisms into operations on the \emph{Turing object} $A$, via the application $\bullet$. Indeed one recovers, as an example, the category $\mathbf{Rec}$, of natural numbers $n, m \in \mathbf{N}$ and partial recursive functions $\mathbf{N}^n \rightarrow \mathbf{N}^m$. This has universal applicative structure:

$\backslash$begin\{center\} $\backslash$begin\{tikzcd\} $\backslash$arrowd, swap, dashed, ``$\backslash$exists $\backslash$hspace\{0.1cm\} $\backslash$text\{total\} $\backslash$hspace\{0.1cm\} h $\backslash$hspace\{0.1cm\} : $\backslash$hspace\{0.1cm\} 1 $\backslash$times h'' $\backslash$mathbf\{N\} $\backslash$times 1 $\backslash$simeq $\backslash$mathbf\{N\} $\backslash$arrowrd, ``f\_i'' $\backslash$ $\backslash$mathbf\{N\} $\backslash$times $\backslash$mathbf\{N\} $\backslash$arrowr, swap, ``$\backslash$langle -- $\backslash$rangle'' \& $\backslash$mathbf\{N\} $\backslash$end\{tikzcd\} $\backslash$end\{center\}

and universal representation $\langle i, n \rangle = f_i(n)$ of the $i$th computable function. In this case the PCA is [[Kleene's first algebra]]. More generally, the computable map category of any PCA forms a Turing category, with Turing object the PCA and its Turing morphism being the applicative structure.

And conversely, a Turing object $A$ of $\mathcal{C}$ and its Turing morphism $\bullet : A \times A \rightarrow A$ form a (relative) PCA. This is effectively a PCA-object internal to an appropriate category:

\begin{defn}
\label{RelativePCA}\hypertarget{RelativePCA}{}
\textbf{(Relative PCA)}

A (relative) PCA $A$ is a combinatory complete partial applicative system in a cartesian [[restriction category]] $\mathcal{D}$, i.e. a morphism $\bullet : A \times A \rightarrow A$ in $\mathcal{D}$ such that finite powers $A^n$ and $A$-computable morphisms form a well-defined cartesian restriction subcategory of $\mathcal{D}$.

\end{defn}
Note, however, that not every relative PCA in a Turing category $\mathcal{C}$ is a Turing object for $\mathcal{C}$.

\hypertarget{basic_properties}{}\subsubsection*{{Basic properties}}\label{basic_properties}

The arithmetical representation of a (partial) function $f : \mathbf{N} \rightarrow \mathbf{N}$ as a member of the recursive enumeration of relations $\varphi_i \subset \mathbf{N} \times \mathbf{N}$ is universal, and this representation satisfies the Kleene [[recursion theorems]]:

$\bullet$ (smn) Let $f: \mathbf{N} \rightarrow \mathbf{N}$ be a partial computable function. Then there is a partial computable function $s : \mathbf{N} \rightarrow \mathbf{N}$ such that $f(\langle i, n \rangle) = f_{s(i)} (n)$.

$\bullet$ (utm) There is a partial computable function $U : \mathbf{N} \rightarrow \mathbf{N}$ such that $U(\langle i, n \rangle ) = f_i (n)$.

As can be nearly read off the definition above, these properties hold with respect to any Turing object. This is part of the overall motivation for Turing categories, since they provide a more model-independent setting both for stating the main theorems of recursion theory, while also allowing the ``intrinsic meaning'' of different constructions in computability theory to stand out. For example, in [[Type II computability]] (or [[function realizability]]) it turns out that \emph{function computability} with respect to the PCA $\mathbf{N}^\mathbf{N}$ (either as [[Kleene's second algebra]] or the [[Baire space of sequences]]) has the topological meaning of \emph{continuity}. Hence, despite the universal availability of Gödel-encodings of all ``$A$-computations'' for an appropriate data type $A$ into $\mathbf{N}$-computations (a form of the [[Church-Turing thesis]]), the theory of Turing categories and their applications to categorical realizability models, for example, are intended to give the intrinsic meaning of computational mathematics without resorting to such a universal representation, much like the success story of Type II computability.

\hypertarget{more_examples}{}\subsection*{{More Examples}}\label{more_examples}

(\ldots{})

\hypertarget{weak_limits_and_exact_completions}{}\subsection*{{Weak Limits and Exact Completions}}\label{weak_limits_and_exact_completions}

(\ldots{})

\hypertarget{references}{}\subsection*{{References}}\label{references}

[[!redirects turing category]] [[!redirects turing-categories]] [[!redirects Turing category]] [[!redirects turing categories]]



\end{document}