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 locales, 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.

This duality may be realized via a dualizing object as follows. The two-element Boolean algebra may be regarded as a Boolean algebra object $\mathbf{2}$ internal to the category of compact Hausdorff spaces $CH$. Thus, for each finitary Boolean algebra operation $\theta\colon \mathbf{2}^n \to \mathbf{2}$, there is a corresponding operation on the representable functor $CH(-, \mathbf{2}): CH^{op} \to Set$ given by

$$CH(-, \mathbf{2})^n \cong CH(-, \mathbf{2}^n) \stackrel{CH(-, \theta)}{\to} CH(-, \mathbf{2})$$

and therefore we obtain a lift

$$CH(-, \mathbf{2}): CH^{op} \to Bool$$

A Stone space is by definition a totally disconnected compact Hausdorff topological space. Let $Stone \hookrightarrow CH$ denote the full subcategory of Stone spaces.

Theorem (Stone representation)

The representable functor restricts to an equivalence of categories $Stone^{op} \to Bool$.

This important theorem can be exploited to give a third description of the free Boolean algebra on a set $X$:

$$Bool(X) \cong CH(2^X, \mathbf{2})$$

where $2$ denotes the 2-element compact Hausdorff space, and $2^X$ the product space $\prod_X 2$. Indeed, the inverse equivalence

$$Bool^{op} \to Stone$$

takes a Boolean algebra $B$ to its spectrum, i.e., the space of Boolean algebra maps $Bool(B, 2)$ (this $2$ is the two-element Boolean algebra $\mathbb{Z}_2$!) equipped with the Zariski topology. Applied to $B = Bool(X)$, we have

$$Bool(B, 2) \cong Set(X, 2) = 2^X$$

where the Zariski topology coincides with the product topology on $2^X$. By the equivalence, we therefore retrieve $Bool(X)$ as $CH(2^X, \mathbf{2})$. This in turn is identified with the Boolean algebra of clopen subsets of the generalised Cantor space $2^X$.

A second description of the inverse equivalence $Bool^{op} \to Stone$ comes about through the yoga of ambimorphic objects. Namely, the Boolean compact Hausdorff space $\mathbf{2}$ can equally well be seen as a compact Hausdorff object in the category of Boolean algebras. Thus, the representable functor $Bool(-, \mathbf{2}): Bool^{op} \to Set$ lifts canonically to a functor

$$Bool^{op} \to CH$$

and in fact part of the Stone representation theorem is that this factors through the inclusion $Stone \hookrightarrow CH$ as the inverse equivalence $Bool^{op} \to Stone$. In particular this lift determines the topology, providing an description alternative to the description in terms of the Zariski topology (although they are of course the same).

An extension of the classical Stone duality to the category of Boolean spaces (= zero-dimensional locally compact Hausdorff spaces) and continuous maps (respectively, perfect maps) was obtained by G. D. Dimov (respectively, by H. P. Doctor) (see the references below).

Stone spaces and profinite sets

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

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

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

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

Stonean spaces and complete Boolean algebras

The category of Stonean spaces, i.e., compact extremally disconnected Hausdorff topological spaces equipped with open continuous maps as morphisms, is contravariantly equivalent to the category of complete Boolean algebras and continuous Boolean homomorphisms as morphisms.

See complete Boolean algebra for more information.

Profinite algebras

If $T$ is a Lawvere theory on $Set$, we can talk about Stone $T$-algebras, i.e. $T$-algebras with a compatible Stone topology, and compare the resulting category $T Alg(Stone)$ with the category $Pro(Fin T Alg)$ 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. $Gpd(Stone)$ is not equivalent to $Pro(FinGpd)$, in contrast to the case for groups. (Of course, groupoids are not described by a Lawvere theory.)

Related concepts

• canonical extension

• syntax - semantics duality

• abstract Stone duality

• Gelfand duality

• Isbell duality

• Stone gamut

References

• Peter Johnstone, Stone Spaces

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

• G. D. Dimov, Some generalizations of the Stone Duality Theorem, Publ. Math. Debrecen 80/3-4 (2012), 255–293.

• G. Dimov, E. Ivanova-Dimova, W. Tholen, Categorical extension of dualities: From Stone to de Vries and beyond, I. Appl. Categ. Struct. 30, 287–329 (2022) doi

• H. P. Doctor, The categories of Boolean lattices, Boolean rings and Boolean spaces, Canad. Math. Bulletin 7 (1964), 245–252 doi

• Jacob Lurie, section A.1.1 of Spectral Algebraic Geometry.

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

Other variants are in

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

• Spencer Breiner, Scheme representation for first-order logic, Ph.D. thesis, Carnegie Mellon U. 2014, pdf

Discussion in E-∞ geometry is in

• Jacob Lurie, section A of Proper Morphisms, Completions, and the Grothendieck Existence Theorem

Discussion of an $(\infty, 1)$-version of Stone duality is in

• Jacob Lurie, section 3.5.3 of Spectral Algebraic Geometry