# MoL-2006-09: Connecting the profinite completion and the canonical extension using duality

MoL-2006-09: Vosmaer, Jacob (2006) Connecting the profinite completion and the canonical extension using duality. [Report]

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We show using duality and category theory that the profinite completion $\mathbb{\hat A}$ of a bounded distributive lattice expansion $\mathbb{A}$ is a homomorphic image of the canonical extension $\mathbb{A}^{\sigma}$. Moreover the natural mapping $\mu\colon \mathbb{A} \rightarrow \mathbb{\hat A}$ can be extended to a surjection $\nu \colon \mathbb{A}^{\sigma} \twoheadrightarrow \mathbb{\hat A}$.