nLab
realizability topos

Context

Topos Theory

topos theory

Background

Toposes

Internal Logic

Topos morphisms

Extra stuff, structure, properties

Cohomology and homotopy

In higher category theory

Theorems

Constructivism, Realizability, Computability

Contents

Idea

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 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.

Constructions

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

Via tripos theory

Via assemblies

Definition

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.

Proposition

Pass APass_A is lextensive.

Theorem

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

Remark

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).

Properties

Axiomatic characterization

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

Theorem

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)

References

  • Stijn Vermeeren, Realizability Toposes, 2009 (pdf)

  • Matías Menni, Exact completions and toposes. Ph.D. Thesis, University of Edinburgh (2000). (web)

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Revised on March 14, 2015 23:55:37 by Bas Spitters (108.17.80.235)