constructive mathematics, realizability, computability
propositions as types, proofs as programs, computational trinitarianism
A realizability topos is a topos which embodies the realizability interpretation of 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.
When the PCA is Kleene's first algebra $\mathcal{K}_1$, the resulting topos is called the effective topos $RT(\mathcal{K}_1)$.
When the PCA is Kleene's second algebra $\mathcal{K}_2$ then $RT(\mathcal{K}_2)$ is the function realizability topos.
There are a number of approaches toward constructing realizability toposes. One is through tripos theory, and another is through assemblies.
…
Let $A$ be a PCA. An ($A$-)partitioned assembly $X$ consists of a set ${|X|}$ and a function $[-]_X \colon {|X|} \to A$. A morphism $X \to Y$ between partitioned assemblies is a function $f \colon {|X|} \to {|Y|}$ for which there exists $a \in A$ such that $a[x]_X$ is defined for all $x \in X$ and $a[x]_X = [f(x)]_Y$. The category of partitioned assemblies is denoted $Pass_A$.
$Pass_A$ is lextensive.
The ex/lex completion of $Pass_A$ is a topos, called the realizability topos of $A$.
A general result about the ex/lex completion $C_{ex/lex}$ of a left exact category $C$ is that it has enough regular 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 this result).
The following is a statement characterizing realizability toposes which is analogous to the Giraud axioms characterizing Grothendieck toposes.
A locally small category $\mathcal{E}$ is (equivalent to) a realizability topos precisely if
$\mathcal{E}$ is exact and locally cartesian closed;
$\mathcal{E}$ has enough projectives and the full subcategory $Proj(\mathcal{E}) \hookrightarrow \mathcal{E}$ has all finite limits;
the global section functor $\Gamma \coloneqq \mathcal{E}(\ast,-) \colon \mathcal{E}\longrightarrow$ Set
has a right adjoint $\nabla \colon Set \hookrightarrow \mathcal{E}$ (which is necessarily a reflective inclusion making $\nabla \Gamma$ a finite-limit preserving idempotent monad/closure operator);
$\nabla$ factors through $Proj(\mathcal{E})$;
there exists an object $D \in Proj(\mathcal{E})$ such that
$D$ is $\nabla\Gamma$-separated (in that its $(\Gamma \dashv \nabla)$-unit is a monomorphism);
all $\nabla \Gamma$-closed regular epimorphisms have the left lifting property against $D\to \ast$;
for every projective object $P$ there is a $\nabla \Gamma$-closed morphism $P \to D$.
This is due to (Frey 14)
Stijn Vermeeren, Realizability Toposes, 2009 (pdf)
Matías Menni, Exact completions and toposes. Ph.D. Thesis, University of Edinburgh (2000). (web)
A characterization of realizability toposes analogous to the Giraud axioms for Grothendieck toposes is given in