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A PREDICATIVE VARIANT OF HYLAND’S EFFECTIVE TOPOS

Published online by Cambridge University Press:  22 October 2020

MARIA EMILIA MAIETTI
Affiliation:
DIPARTIMENTO DI MATEMATICA “TULLIO LEVI-CIVITA” UNIVERSITÀ DI PADOVAPADOVA, ITALYE-mail:maietti@math.unipd.itE-mail:maschio@math.unipd.it
SAMUELE MASCHIO
Affiliation:
DIPARTIMENTO DI MATEMATICA “TULLIO LEVI-CIVITA” UNIVERSITÀ DI PADOVAPADOVA, ITALYE-mail:maietti@math.unipd.itE-mail:maschio@math.unipd.it

Abstract

Here, we present a category ${\mathbf {pEff}}$ which can be considered a predicative variant of Hyland's Effective Topos ${{\mathbf {Eff} }}$ for the following reasons. First, its construction is carried in Feferman’s predicative theory of non-iterative fixpoints ${{\widehat {ID_1}}}$ . Second, ${\mathbf {pEff}}$ is a list-arithmetic locally cartesian closed pretopos with a full subcategory ${{\mathbf {pEff}_{set}}}$ of small objects having the same categorical structure which is preserved by the embedding in ${\mathbf {pEff}}$ ; furthermore subobjects in ${{\mathbf {pEff}_{set}}}$ are classified by a non-small object in ${\mathbf {pEff}}$ . Third ${\mathbf {pEff}}$ happens to coincide with the exact completion of the lex category defined as a predicative rendering in ${{\widehat {ID_1}}}$ of the subcategory of ${{\mathbf {Eff} }}$ of recursive functions and it validates the Formal Church’s thesis. Hence pEff turns out to be itself a predicative rendering of a full subcategory of ${{\mathbf {Eff} }}$ .

Type
Article
Copyright
© The Association for Symbolic Logic 2020

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