A sheaf representation and duality for finitely presented Heyting algebras

Journal of Symbolic Logic 60 (3):911-939 (1995)
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Abstract

A. M. Pitts in [Pi] proved that HA op fp is a bi-Heyting category satisfying the Lawrence condition. We show that the embedding $\Phi: HA^\mathrm{op}_\mathrm{fp} \longrightarrow Sh(\mathbf{P_0,J_0})$ into the topos of sheaves, (P 0 is the category of finite rooted posets and open maps, J 0 the canonical topology on P 0 ) given by $H \longmapsto HA(H,\mathscr{D}(-)): \mathbf{P_0} \longrightarrow \text{Set}$ preserves the structure mentioned above, finite coproducts, and subobject classifier, it is also conservative. This whole structure on HA op fp can be derived from that of Sh(P 0 ,J 0 ) via the embedding Φ. We also show that the equivalence relations in HA op fp are not effective in general. On the way to these results we establish a new kind of duality between HA op fp 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]

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Citations of this work

Rules and Arithmetics.Albert Visser - 1999 - Notre Dame Journal of Formal Logic 40 (1):116-140.
An algebraic theory of normal forms.Silvio Ghilardi - 1995 - Annals of Pure and Applied Logic 71 (3):189-245.
Pitts' Quantifiers Are Not Topological Quantification.Tomasz Połacik - 1998 - Notre Dame Journal of Formal Logic 39 (4):531-544.

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References found in this work

An algebraic theory of normal forms.Silvio Ghilardi - 1995 - Annals of Pure and Applied Logic 71 (3):189-245.
Characters and fixed-points in provability logic.Zachary Gleit & Warren Goldfarb - 1989 - Notre Dame Journal of Formal Logic 31 (1):26-36.
Descent and duality.Marek W. Zawadowski - 1995 - Annals of Pure and Applied Logic 71 (2):131-188.

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