realizability topos



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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.


There are a number of approaches toward constructing realizability toposes. One is through tripos theory, and another is through assemblies.

Via tripos theory

See tripos.

Via assemblies


Let AA be a PCA. An (AA-)partitioned assembly XX consists of a set |X|{|X|} and a function [] X:|X|A[-]_X \colon {|X|} \to A. A morphism XYX \to Y between partitioned assemblies is a function f:|X||Y|f \colon {|X|} \to {|Y|} for which there exists aAa \in A such that a[x] Xa[x]_X is defined for all xXx \in X and a[x] X=[f(x)] Ya[x]_X = [f(x)]_Y. The category of partitioned assemblies is denoted Pass APass_A.


Pass APass_A is lextensive.


The ex/lex completion of Pass APass_A is a topos, called the realizability topos of AA.


A general result about the ex/lex completion C ex/lexC_{ex/lex} of a left exact category CC is that it has enough regular projectives, meaning objects PP such that hom(P,):C ex/lexSet\hom(P, -) \colon C_{ex/lex} \to Set preserves regular epis. In fact, the regular projective objects coincide with the objects of CC (as a subcategory of C ex/lexC_{ex/lex}). Of course, when C ex/lexC_{ex/lex} is a topos, where every epi is regular, this means C ex/lexC_{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 CC, are again objects of CC (see this result).


Axiomatic characterization

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

  1. \mathcal{E} is exact and locally cartesian closed;

  2. \mathcal{E} has enough projectives and the full subcategory Proj()Proj(\mathcal{E}) \hookrightarrow \mathcal{E} has all finite limits;

  3. the global section functor Γ(*,):\Gamma \coloneqq \mathcal{E}(\ast,-) \colon \mathcal{E}\longrightarrow Set

    1. has a right adjoint :Set\nabla \colon Set \hookrightarrow \mathcal{E} (which is necessarily a reflective inclusion making Γ\nabla \Gamma a finite-limit preserving idempotent monad/closure operator);

    2. \nabla factors through Proj()Proj(\mathcal{E});

  4. there exists an object DProj()D \in Proj(\mathcal{E}) such that

    1. DD is Γ\nabla\Gamma-separated (in that its (Γ)(\Gamma \dashv \nabla)-unit is a monomorphism);

    2. all Γ\nabla \Gamma-closed regular epimorphisms have the left lifting property against D*D\to \ast;

    3. for every projective object PP there is a Γ\nabla \Gamma-closed morphism PDP \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

Last revised on May 4, 2019 at 08:59:58. See the history of this page for a list of all contributions to it.