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\section*{realizability topos}

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

\hypertarget{topos_theory}{}\paragraph*{{Topos Theory}}\label{topos_theory}

[[!include topos theory - contents]]

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

[[!include constructivism - contents]]

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

\noindent\hyperlink{idea}{Idea}\dotfill \pageref*{idea} \linebreak
\noindent\hyperlink{constructions}{Constructions}\dotfill \pageref*{constructions} \linebreak
\noindent\hyperlink{via_tripos_theory}{Via tripos theory}\dotfill \pageref*{via_tripos_theory} \linebreak
\noindent\hyperlink{via_assemblies}{Via assemblies}\dotfill \pageref*{via_assemblies} \linebreak
\noindent\hyperlink{properties}{Properties}\dotfill \pageref*{properties} \linebreak
\noindent\hyperlink{Characterization}{Axiomatic characterization}\dotfill \pageref*{Characterization} \linebreak
\noindent\hyperlink{related_concepts}{Related concepts}\dotfill \pageref*{related_concepts} \linebreak
\noindent\hyperlink{references}{References}\dotfill \pageref*{references} \linebreak


\hypertarget{idea}{}\subsection*{{Idea}}\label{idea}

A realizability topos is a [[topos]] which embodies the [[realizability interpretation]] of [[intuitionistic mathematics|intuitionistic]] [[number theory]] (due to Kleene) as part of its [[internal logic]]. Realizability toposes form an important class of [[elementary toposes]] that are not [[Grothendieck toposes]], and don't even have a [[geometric morphism]] to [[Set]].

The input datum for forming a realizability topos is a [[partial combinatory algebra]], or PCA.

\begin{itemize}%
\item When the PCA is [[Kleene's first algebra]] $\mathcal{K}_1$, the resulting topos is called the [[effective topos]] $RT(\mathcal{K}_1)$.


\item When the PCA is [[Kleene's second algebra]] $\mathcal{K}_2$ then $RT(\mathcal{K}_2)$ is the [[function realizability topos]].



\end{itemize}
\hypertarget{constructions}{}\subsection*{{Constructions}}\label{constructions}

There are a number of approaches toward constructing realizability toposes. One is through [[tripos]] theory, and another is through assemblies (actually the latter is a family of related approaches).

Let $A$ be a [[partial combinatory algebra|PCA]] --- in [[Set]], for simplicity, but similar constructions usually work over other base toposes.

\hypertarget{via_tripos_theory}{}\subsubsection*{{Via tripos theory}}\label{via_tripos_theory}

There is a [[tripos]] whose base category is $Set$ and for which the preorder $P_A(X)$ of $X$-indexed predicates is the set $P(A)^X$ of functions from $X$ to the [[powerset]] $P(A)$ of $A$. The order relation sets $\phi \le \psi$ if there exists $a\in A$ such that $b\in \phi(x)$ implies $a\cdot b \in \psi(x)$ for all $x$; note that $a$ must be chosen uniformly across all $x\in X$.

Applying the tripos-to-topos construction to this tripos produces the realizability topos over $A$. See [[tripos]] for details.

\hypertarget{via_assemblies}{}\subsubsection*{{Via assemblies}}\label{via_assemblies}

\begin{defn}
\label{}\hypertarget{}{}
An \textbf{assembly} $X$ consists of a set ${|X|}$ and a function $[-]_X \colon {|X|} \to P(A)$, where $P(A)$ denotes the [[powerset]] of $A$. An assembly is \textbf{partitioned} if $[-]_X$ takes values in singletons, i.e. is a function ${|X|} \to A$.

A \textbf{morphism} $X \to Y$ between assemblies is a function $f \colon {|X|} \to {|Y|}$ for which there exists $a \in A$ such that for all $x\in X$ and $b\in [x]_X$, $a\cdot b$ is defined and belongs to $[f(x)]_Y$.

\end{defn}
The categories of assemblies and partitioned assemblies are denoted $Ass_A$ and $PAss_A$ respectively.

\begin{prop}
\label{}\hypertarget{}{}
$Ass_A$ and $PAss_A$ are finitary [[extensive category|lextensive]]. Moreover, $Ass_A$ is [[regular category|regular]] and [[locally cartesian closed category|locally cartesian closed]].

\end{prop}
\begin{defn}
\label{}\hypertarget{}{}
$Ass_A$ is the [[reg/lex completion]] of $PAss_A$. Therefore, [[ex/lex completion]] of $PAss_A$ coincides with the [[ex/reg completion]] of $Ass_A$. This category is a [[topos]], called the \textbf{realizability topos} of $A$.

\end{defn}
\begin{remark}
\label{pax}\hypertarget{pax}{}
A general result about the ex/lex completion $C_{ex/lex}$ of a left exact category $C$ is that it has enough regular [[projective object|projectives]], meaning objects $P$ such that $\hom(P, -) \colon C_{ex/lex} \to Set$ preserves regular epis. In fact, the regular projective objects coincide with the objects of $C$ (as a subcategory of $C_{ex/lex}$). Of course, when $C_{ex/lex}$ is a topos, where every epi is regular, this means $C_{ex/lex}$ has enough projectives, or satisfies (external) [[COSHEP]]. It also satisfies internal [[COSHEP]], since binary products of projectives, i.e., products of objects of $C$, are again objects of $C$ (see \href{/nlab/show/internally+projective+object#enough}{this result}).

\end{remark}
\begin{remark}
\label{ac}\hypertarget{ac}{}
The fact that a realizability topos is an ex/lex completion depends on the [[axiom of choice]] for [[Set]], since it requires the partitioned assemblies to be projective objects therein. In the absence of the axiom of choice, the projective objects in a realizability topos are the (isomorphs of) partitioned assemblies whose underlying set is [[projective object|projective]] in [[Set]]. Thus, if [[COSHEP]] holds in [[Set]], then a realizability topos is the ex/wlex completion of the category of such ``projective partitioned assemblies'' (wlex because this category may not have finite limits, only weak finite limits). Without some choice principle, the realizability topos may not be an ex/wlex completion at all; but it is still an ex/reg completion of $Ass_A$.

\end{remark}
\hypertarget{properties}{}\subsection*{{Properties}}\label{properties}

\hypertarget{Characterization}{}\subsubsection*{{Axiomatic characterization}}\label{Characterization}

The following is a statement characterizing realizability toposes which is analogous to the [[Giraud axioms]] characterizing [[Grothendieck toposes]].

\begin{theorem}
\label{}\hypertarget{}{}
A [[locally small category]] $\mathcal{E}$ is ([[equivalence of categories|equivalent]] to) a realizability topos precisely if

\begin{enumerate}%
\item $\mathcal{E}$ is [[exact category|exact]] and [[locally cartesian closed category|locally cartesian closed]];


\item $\mathcal{E}$ has [[enough projectives]] and the [[full subcategory]] $Proj(\mathcal{E}) \hookrightarrow \mathcal{E}$ has all [[finite limits]];


\item the [[global section]] [[functor]] $\Gamma \coloneqq \mathcal{E}(\ast,-) \colon \mathcal{E}\longrightarrow$ [[Set]]

\begin{enumerate}%
\item has a [[right adjoint]] $\nabla \colon Set \hookrightarrow \mathcal{E}$ (which is necessarily a [[reflective subcategory|reflective inclusion]] making $\nabla \Gamma$ a finite-limit preserving [[idempotent monad]]/[[closure operator]]);


\item $\nabla$ factors through $Proj(\mathcal{E})$;



\end{enumerate}

\item there exists an object $D \in Proj(\mathcal{E})$ such that

\begin{enumerate}%
\item $D$ is $\nabla\Gamma$-[[separated presheaf|separated]] (in that its $(\Gamma \dashv \nabla)$-[[unit of a monad|unit]] is a [[monomorphism]]);


\item all $\nabla \Gamma$-\href{closure+operator#InducedClosureOnSlices}{closed} [[regular epimorphisms]] have the [[left lifting property]] against $D\to \ast$;


\item for every [[projective object]] $P$ there is a $\nabla \Gamma$-\href{closure+operator#InducedClosureOnSlices}{closed morphism} $P \to D$.



\end{enumerate}


\end{enumerate}
\end{theorem}
This is due to (\hyperlink{Frey14}{Frey 14})

\hypertarget{related_concepts}{}\subsection*{{Related concepts}}\label{related_concepts}

\begin{itemize}%
\item [[realizability]]


\item [[realizability model]]



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

\begin{itemize}%
\item [[Stijn Vermeeren]], \emph{Realizability Toposes}, 2009 (\href{http://stijnvermeeren.be/download/mathematics/essay.pdf}{pdf})


\item Mat\'i{}as Menni, Exact completions and toposes. Ph.D. Thesis, University of Edinburgh (2000). (\href{http://www.lfcs.inf.ed.ac.uk/reports/00/ECS-LFCS-00-424/}{web})



\end{itemize}
A characterization of realizability toposes analogous to the [[Giraud axioms]] for [[Grothendieck toposes]] is given in

\begin{itemize}%
\item [[Jonas Frey]], \emph{Characterizing partitioned assemblies and realizability toposes} (\href{http://arxiv.org/abs/1404.6997}{arXiv:1404.6997})

\end{itemize}
[[!redirects realizability topos]] [[!redirects realizability topoi]] [[!redirects realizability toposes]] [[!redirects realisability topos]] [[!redirects realisability topoi]] [[!redirects realisability toposes]]

[[!redirects Realizability topos]] [[!redirects Realizability topoi]] [[!redirects Realizability toposes]] [[!redirects Realisability topos]] [[!redirects Realisability topoi]] [[!redirects Realisability toposes]]

[[!redirects assembly]] [[!redirects assemblies]]



\end{document}