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A sheaf representation and duality for finitely presented Heyting algebras

Published online by Cambridge University Press:  12 March 2014

Silvio Ghilardi  and
Marek Zawadowski
Silvio Ghilardi
Affiliation:
Dipartimento di Matematica, Università degli Studi di Milano, Via C. Saldini 50, 20133 Milano, Italy, E-mail: ghilardi@vmimat.mat.unimi.it
Marek Zawadowski
Affiliation:
Instytut Matematyki, Uniwersytet Warszawski, UL. S.Banacha 2, 00-913 Warszawa, Poland, E-mail: zawado@mimuw.edu.pl
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Abstract

A. M.Pitts in [Pi] proved that is a bi-Heyting category satisfying the Lawvere condition. We show that the embedding Φ: → Sh(P0, J0) into the topos of sheaves, (P0 is the category of finite rooted posets and open maps, J0 the canonical topology on P0) given by H ↦ HA(H, (−)) : P0 → Set preserves the structure mentioned above, finite coproducts, and subobject classifier; it is also conservative. This whole structure on can be derived from that of Sh(P0, J0) via the embedding Φ. We also show that the equivalence relations in are not effective in general. On the way to these results we establish a new kind of duality between and a category of sheaves equipped with certain structure defined in terms of Ehrenfeucht games. Our methods are model-theoretic and combinatorial as opposed to proof-theoretic as in [Pi].

Type
Research Article
Information
The Journal of Symbolic Logic , Volume 60 , Issue 3 , September 1995 , pp. 911 - 939
Copyright
Copyright © Association for Symbolic Logic 1995

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