abstract duality: opposite category,
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.
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.
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.
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 internal to the category of compact Hausdorff spaces . Thus, for each finitary Boolean algebra operation , there is a corresponding operation on the representable functor given by
and therefore we obtain a lift
The representable functor restricts to an equivalence of categories .
This important theorem can be exploited to give a third description of the free Boolean algebra on a set :
where denotes the 2-element compact Hausdorff space, and the product space . Indeed, the inverse equivalence
where the Zariski topology coincides with the product topology on . By the equivalence, we therefore retrieve as . This in turn is identified with the Boolean algebra of clopen subsets of the generalised Cantor space .
A second description of the inverse equivalence comes about through the yoga of ambimorphic objects. Namely, the Boolean compact Hausdorff space can equally well be seen as a compact Hausdorff object in the category of Boolean algebras. Thus, the representable functor lifts canonically to a functor
and in fact part of the Stone representation theorem is that this factors through the inclusion as the inverse equivalence . 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).
Note that a finite Stone space is necessarily discrete, and these correspond to the finite Boolean algebras, i.e. . However, since Boolean algebras form a locally finitely presentable category, we have (see ind-object and pro-object). In consequence, : 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
If is a Lawvere theory on , we can talk about Stone -algebras, i.e. -algebras with a compatible Stone topology, and compare the resulting category with the category of pro-(finite -algebras). The previous duality says that these categories are equivalent when is the identity theory. It is also true in many other cases, such as:
However it is false for some , such as:
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. is not equivalent to , in contrast to the case for groups. (Of course, groupoids are not described by a Lawvere theory.)
G. D. Dimov, Some generalizations of the Stone Duality Theorem, Publ. Math. Debrecen 80/3-4 (2012), 255–293.
H. P. Doctor, The categories of Boolean lattices, Boolean rings and Boolean spaces, Canad. Math. Bulletin 7 (1964), 245–252.
There is a version in model theory, Makkai duality,
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
Discussion of an -version of Stone duality is in