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  "body": "\n## Definition\n\nLet $\\mathcal{A}$ be a [[variety|AJNZJD8442]] or a [[quasivariety|WRXEKT4NQS]] of algebras, regarded as a category whose morphisms are the homomorphisms.\nLet $\\mathcal{X}$ be a category whose objects are [[Stone spaces|ZV38HA5PAQ]] with additional relations or operations.\nRecall that a Stone space is a compact Hausdorff totally disconnected space.\nThe morphisms of $\\mathcal{X}$ are the continuous maps that preserve the additional structure.\nA *Stone-type duality* for $\\mathcal{A}$ is an [[equivalence|K7A04ZBD92]] between $\\mathcal{A}$ and the [[opposite category|DAMK8XE1FR]] $\\mathcal{X}^{\\mathrm{op}}$.\nEquivalently, it is a pair of contravariant functors $D \\colon \\mathcal{A} \\to \\mathcal{X}$ and $E \\colon \\mathcal{X} \\to \\mathcal{A}$ with natural isomorphisms $A \\cong E(D(A))$ and $X \\cong D(E(X))$.\n\n## Remarks\n\nIn most examples $D$ and $E$ are represented by one set with two structures.\nThen $D(A)$ is the set of homomorphisms from $A$ to a fixed finite algebra.\nAnd $E(X)$ is the set of morphisms from $X$ to the same set with a discrete topology.\n\nExample: Boolean algebras and Stone spaces.\nHere $D(A)$ is the set of homomorphisms $A \\to \\set{0, 1}$, a closed subspace of $\\set{0, 1}^A$, and $E(X)$ is the Boolean algebra of clopen subsets of $X$.\nBoth composites are naturally isomorphic to the identity, by Stone's representation theorem.\n\nNon-example: Pontryagin duality between abelian groups and compact abelian groups is a dual equivalence, but not of this type.\nThe circle group $\\R / \\Z$ is the dual of $\\Z$, and it is connected, so it is not a Stone space.\n\nBounded distributive lattices are dual to Priestley spaces, which are Stone spaces with a partial order satisfying a separation condition.\n",
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