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 (actually the latter is a family of related approaches).
Let $A$ be a PCA — in Set, for simplicity, but similar constructions usually work over other base toposes.
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.
An assembly $X$ consists of a set ${|X|}$ and a function $[-]_X \colon {|X|} \to P_{\ge 1}(A)$, where $P_{\ge 1}(A)$ denotes the set of inhabited subsets of $A$. An assembly is partitioned if $[-]_X$ takes values in singletons, i.e. is a function ${|X|} \to A$.
A 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 $[f(x)]_Y$.
The categories of assemblies and partitioned assemblies are denoted $Ass_A$ and $PAss_A$ respectively.
$Ass_A$ and $PAss_A$ are finitary lextensive. Moreover, $Ass_A$ is regular and locally cartesian closed.
$Ass_A$ 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 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 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 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 $Ass_A$.
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