Stone-type duality for varieties

Builds on Variety of algebras Quasivariety Stone space Equivalence of categories Opposite category

Definition

Let $\mathcal{A}$ be a variety or a quasivariety of algebras, regarded as a category whose morphisms are the homomorphisms. Let $\mathcal{X}$ be a category whose objects are Stone spaces with additional relations or operations. Recall that a Stone space is a compact Hausdorff totally disconnected space. The morphisms of $\mathcal{X}$ are the continuous maps that preserve the additional structure. A Stone-type duality for $\mathcal{A}$ is an equivalence between $\mathcal{A}$ and the opposite category $\mathcal{X}^{\mathrm{op}}$ . Equivalently, 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 $X \cong D(E(X))$ .

Remarks

In most examples $D$ and $E$ are represented by one set with two structures. Then $D(A)$ is the set of homomorphisms from $A$ to a fixed finite algebra. And $E(X)$ is the set of morphisms from $X$ to the same set with a discrete topology.

Example: Boolean algebras and Stone spaces. Here $D(A)$ 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 $X$ . Both composites are naturally isomorphic to the identity, by Stone's representation theorem.

Non-example: Pontryagin duality between abelian groups and compact abelian groups is a dual equivalence, but not of this type. The circle group $\R / \Z$ is the dual of $\Z$ , and it is connected, so it is not a Stone space.

Bounded distributive lattices are dual to Priestley spaces, which are Stone spaces with a partial order satisfying a separation condition.

Reading path

To understand this page from the ground up, read these 6 pages in order. The foundations come first.

Show the path as a graph

Each page uses the pages above it. The colour of the bar on a box shows how strong the evidence is, and the box also gives it as text. The list has the same pages.

no evidence level

Equivalence of categories concept
Opposite category concept
Quasivariety concept
Stone space concept
Variety of algebras concept
Stone-type duality for varieties definition

Tag	`V8PEAAE4K1`
URI	`m2://V8PEAAE4K1`
Kind	definition
Writing	W0 stub
MSC	08C20, 06D50, 06E15
Defines	Stone-type duality for varieties
Uses	Variety of algebras, Quasivariety, Stone space, Equivalence of categories, Opposite category

Stone-type duality for varieties

Definition

Remarks

Reading path

Further reading

What this page needs for W1

What links here