nLab
Stone duality

Idea

Stone duality is a subject comprising various dualities between space and quantity in the area of general topology and topological algebra.

Particular cases

Locales and frames

Perhaps the most general duality falling under this heading is that between locales (on the space side) and frames (on the quantity side). Of course, this duality is not very deep at all; the category Loc of locales is simply defined to be the opposite of the category Frm of frames. But there are several interesting dualities between subcategories of these.

Topological spaces

Stone duality is often described for topological spaces rather than for locales. In this case, the most general duality is that between sober spaces and frames with enough points (which correspond to topological locales). In many cases, one requires the ultrafilter theorem (or other forms of the axiom of choice) in order for the duality to hold when applied to topological spaces, while the duality holds for locales even in constructive mathematics.

Coherent spaces and distributive lattices

Any distributive lattice generates a free frame. The locales which arise in this way can be characterized as the coherent locale?s, and this gives a duality between distributive lattices and coherent locales. Note that one must additionally restrict to “coherent maps” between coherent locales. Also, at least assuming the axiom of choice, every coherent locale is topological, so we may say “coherent space” instead.

Stone spaces and Boolean algebras

The duality which is due to Marshall Stone, and which gives its name to the subject, is the duality between Stone spaces and Boolean algebras. Specifically, a distributive lattice is a Boolean algebra precisely when the free frame it generates is the topology of a Stone space, and any continuous map of Stone spaces is coherent. Therefore, the category of Stone spaces is dual to the category of Boolean algebras. The Boolean algebra corresponding to a Stone space consists of its clopen sets.

One way of explaining this classical Stone duality is via the following sequence of equivalences of categories

$$\mathrm{Bool}\simeq \mathrm{Ind}(\mathrm{FinBool})\simeq \mathrm{Ind}({\mathrm{FinSet}}^{\mathrm{op}})\simeq \mathrm{Pro}(\mathrm{FinSet}{)}^{\mathrm{op}},$$Bool \simeq Ind(FinBool) \simeq Ind(FinSet^{op}) \simeq Pro(FinSet)^{op} \,,

where “FinSet” is the category of finite sets, ”$\mathrm{Ind}$” stands for ind-objects, ”$\mathrm{Pro}$” for pro-objects and ${}^{\mathrm{op}}$ for the opposite category and the equivalence ${\mathrm{FinSet}}^{\mathrm{op}}\simeq \mathrm{FinBool}$ is that discussed at FinSet – Opposite category.

Stone spaces and profinite sets

Note that a finite Stone space is necessarily discrete, and these correspond to the finite Boolean algebras, i.e. $\mathrm{FinSet}\simeq \mathrm{FinStoneTop}\simeq {\mathrm{FinBool}}^{\mathrm{op}}$. However, since Boolean algebras form a locally finitely presentable category, we have $\mathrm{Bool}\simeq \mathrm{Ind}(\mathrm{FinBool})\simeq \mathrm{Pro}(\mathrm{FinSet}{)}^{\mathrm{op}}$ (see ind-object and pro-object). In consequence, $\mathrm{StoneTop}\simeq \mathrm{Pro}(\mathrm{FinSet})$: i.e. Stone spaces are equivalent to profinite sets.

Profinite algebras

If $T$ is a Lawvere theory on $\mathrm{Set}$, we can talk about Stone $T$-algebras, i.e. $T$-algebras with a compatible Stone topology, and compare the resulting category $T\mathrm{Alg}(\mathrm{Stone})$ with the category $\mathrm{Pro}(\mathrm{Fin}T\mathrm{Alg})$ of pro-(finite $T$-algebras). The previous duality says that these categories are equivalent when $T$ is the identity theory. It is also true in many other cases, such as:

• the theory of groups, resulting in the rich theory of profinite groups
• the theories of semigroups and monoids
• the theory of rings (with or without 1)
• the theories of distributive lattices, Heyting algebras, and Boolean algebras
• the theory of $M$-sets, where $M$ is a finite monoid
• the theories of $R$-modules and $R$-algebras, where $R$ is a finite ring

However it is false for some $T$, such as:

• the theory of $\mathbb{N}$-sets, i.e. sets equipped with an endomorphism
• the theory of Jónsson-Tarski algebras
• the theory of lattices

All of these can be found in chapter VI of Johnstone’s book cited below.

The corresponding fact is also notably false for groupoids, i.e. $\mathrm{Gpd}(\mathrm{Stone})$ is not equivalent to $\mathrm{Pro}(\mathrm{FinGpd})$, in contrast to the case for groups. (Of course, groupoids are not described by a Lawvere theory.)

Related concepts

• canonical extension

• syntax - semantics duality

References

The book

• Peter Johnstone, Stone Spaces

is all about Stone duality.

• Olivia Caramello, A topos-theoretic approach to Stone-type dualities, arxiv/1103.3493 158 pp.

There is a version in model theory, Makkai duality,

• M. Makkai, Stone duality for first-order logic, Adv. Math. 65 (1987) no. 2, 97–170, doi, MR89h:03067; Duality and definability in first order logic, Mem. Amer. Math. Soc. 105 (1993), no. 503

Another variant is in

• Henrik Forssell, First-order logical duality, Ph.D. thesis, Carnegie Mellon U. 2008, pdf