Marta Bunge - Relative stone duality

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Marta Bunge - Relative stone duality

MARTA BUNGE, Department of Mathematics and Statistics, University of McGill Montréal, Québec, H3A 2K6

Relative stone duality

Let $e\colon \cal E \to \cal S$ be a bounded geometric morphism between elementary toposes. We prove a relative pure/entire factorization of any geometric morphism over $\cal S$ whose domain is a dominance (subopen and such that $\cal S$ -definable monos in it compose). Closely related to it is a relative Stone Duality. Denote by $\textrm{DL}_{\Omega_{\cal S}}(\cal E)$ the category of $\Omega_{\cal S}$ -distributive lattices in $\cal E$ and $\Omega_{\cal S}$ -action preserving lattice homomorphisms, and by $\textrm{FRM}_{\Omega_{\cal S}}(\cal E)$ the category of frames A in $\cal E$ for which the corresponding topos ${\cal E}[A]$ of sheaves on A is a dominance over $\cal S$ and frame homomorphisms. We prove that there is a duality between these categories and that it restricts to an equivalence between suitably defined categories $\textrm{Boole}_{\Omega_{\cal S}}({\cal E})$ and $\textrm{Stone}_{\Omega_{\cal S}}({\cal E})^{op}$ . When $\mathcal{S}$ is Sets, this reduces to the usual Stone Duality. As an application, we answer a question of P. T. Johnstone (Cartesian monads on toposes, J. Pure Appl. Alg. 116(1997), 199-220). This is part of ongoing work on ``Distribution Algebras'', joint with J. Funk (UBC), M. Jibladze (Louvain-la-Neuve) and T. Streicher (Darmstadt).)

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