nLab
realizability topos

Contents

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 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.

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 AA be a PCA — in Set, for simplicity, but similar constructions usually work over other base toposes.

Via tripos theory

There is a tripos whose base category is SetSet and for which the preorder P A(X)P_A(X) of XX-indexed predicates is the set P(A) XP(A)^X of functions from XX to the powerset P(A)P(A) of AA. The order relation sets ϕψ\phi \le \psi if there exists aAa\in A such that bϕ(x)b\in \phi(x) implies abψ(x)a\cdot b \in \psi(x) for all xx; note that aa must be chosen uniformly across all xXx\in X.

Applying the tripos-to-topos construction to this tripos produces the realizability topos over AA. See tripos for details.

Via assemblies

Definition

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

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

The categories of assemblies and partitioned assemblies are denoted Ass AAss_A and PAss APAss_A respectively.

Proposition

Ass AAss_A and PAss APAss_A are finitary lextensive. Moreover, Ass AAss_A is regular and locally cartesian closed.

Theorem

Ass AAss_A is the reg/lex completion of PAss APAss_A. Therefore, ex/lex completion of PAss APAss_A coincides with the ex/reg completion of Ass AAss_A. This category 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).

Remark

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 AAss_A.

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)

A characterization of realizability toposes analogous to the Giraud axioms for Grothendieck toposes is given in

Last revised on June 16, 2019 at 17:31:07. See the history of this page for a list of all contributions to it.