Geometric and Higher Order Logic in terms of Abstract Stone Duality

Paul Taylor

The contravariant powerset, and its generalisations $\Sigma^X$ to the lattices of open subsets of a locally compact topological space and of recursively enumerable subsets of numbers, satisfy the Euclidean principle that $\phi\meet F(\phi)=\phi\meet F(\top)$.

Conversely, when the adjunction $\Sigma^{(-)}\dashv\Sigma^{(-)}$ is monadic, this equation implies that $\Sigma$ classifies some class of monos, and the Frobenius law $\exists x.(\phi(x)\meet\psi)=(\exists x.\phi(x))\meet\psi)$ for the existential quantifier.

In topology, the lattice duals of these equations also hold, and are related to the Phoa principle in synthetic domain theory.

The natural definitions of discrete and Hausdorff spaces correspond to equality and inequality, whilst the quantifiers considered as adjoints characterise open (or, as we call them, overt) and compact spaces. Our treatment of overt discrete spaces and open maps is precisely dual to that of compact Hausdorff spaces and proper maps.

The category of overt discrete spaces forms a pretopos and the paper concludes with a converse of Paré's theorem (that the contravariant powerset functor is monadic) that characterises elementary toposes by means of the monadic and Euclidean properties together with all quantifiers, making no reference to subsets.

Keywords: locally compact spaces; compact Hausdor. spaces, open spaces; overt spaces; open maps; proper maps; higher order logic; contravariant powerset; elementary topos; subobject classifier; Euclidean principle; synthetic domain theory; monadic adjunction; pretopos; quantifiers.

2000 MSC: 03F35 06D22 06E15 18A15 18B05 18C20 54A05 54C10, 54D10 54D30 54D45.

Theory and Applications of Categories, Vol. 7, 2000, No. 15, pp 284-338.

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