On (co)products of partial combinatory algebras, with an application to pushouts of realizability toposes

Jetze Zoethout

doi:10.1017/s096012952100013x

Abstract

AbstractWe consider two preorder-enriched categories of ordered partial combinatory algebras: OPCA, where the arrows are functional (i.e., projective) morphisms, and OPCA†, where the arrows are applicative morphisms. We show that OPCA has small products and finite biproducts, and that OPCA† has finite coproducts, all in a suitable 2-categorical sense. On the other hand, OPCA† lacks all nontrivial binary products. We deduce from this that the pushout, over Set, of two nontrivial realizability toposes is never a realizability topos. In contrast, we show that nontrivial subtoposes of realizability toposes are closed under pushouts over Set.

Highlights

Intuitionistic mathematics, or more precisely, the BHK interpretation of intuitionistic mathematics, employs a notion of function or construction
The Realizability Topos In this final section, we briefly investigate what we can say about coproducts of the realizability toposes associated to ordered partial combinatory algebras (OPCAs); in particular, to which extent realizability toposes are closed under coproducts
OPCA has a better categorical structure than OPCA†. It seems that OPCA† is more important for the study of functors between categories of assemblies and realizability toposes

Summary

Introduction

Intuitionistic mathematics, or more precisely, the BHK interpretation of intuitionistic mathematics, employs a notion of function or construction. (i) A is equivalent to 1; (ii) A has a least element; (iii) idA is a zero morphism; (iv) ¡ : 1 → A is c.d. An OPCA A satisfying the equivalent conditions of Lemma 3.4 will be called trivial. In Hofstra and Van Oosten (2003), it is shown how to reconstruct the notion of applicative morphism by introducing a certain pseudomonad on OPCA. This is the treatment we follow here. Suppose that f : A B and g : B C are c.d. fand gare c.d., so by Proposition 2.14(i), gf gfis c.d., gf is c.d. there exists a pseudofunctor OPCA → OPCA† sending a morphism f : A → B to δBf : A B. This means that we can speak unambiguously about the equivalence of OPCAs

Products and Coproducts in PCA

Findings

Conclusion