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Through the Looking Glass

In mathematics, a structure may have an alter ego in a completely different world. Categorical duality is the mirror that lets information pass between them—from logic to sets, algebra to geometry, and equations to spaces.

Alter egos in mathematics

Just as Dr Jekyll has Mr Hyde, and Jake Sully has his avatar in the world of the Na’vi, so too in mathematics a structure may possess an alter ego that lives in an entirely different world.

This hall of mirrors is formalised by what mathematicians call categorical duality, which translates information between different worlds. In this article, we will take a short journey through some of these dualities. We will meet strange but inseparable pairs: geometry and algebra, sets and logic, curves and equations. One side identifies points; the other manipulates symbols. Yet they tell the same story.

An example: Stone duality

Our first example is called Stone duality. To introduce it, we must travel back more than 150 years, to the time when George Boole packaged the rules of reasoning into a mathematical object: the Boolean algebra.

A Boolean algebra is a set—whose elements we may think of as propositions, or statements—equipped with several operations:

  • p ∧ q, read “p and q”, which is true precisely when both p and q are true;
  • p ∨ q, read “p or q”, which is true precisely when at least one of p and q is true;
  • ¬p, read “not p”, which is true precisely when p is not.

It also has two special elements, written 1 (true) and 0 (false). An arbitrary proposition may be true or false, but 1 is always true and 0 is always false.

For these ingredients to form a Boolean algebra, the operations must satisfy equations reminiscent of the laws of set theory. For example, the equation p ∧ ¬p = 0—“a proposition and its negation cannot both hold”—echoes the fact that the intersection of a subset P and its complement Pc is empty.

Indeed, the power set P(X), consisting of all subsets of a set X, is a Boolean algebra under intersection, union, and complementation. More generally, any family of subsets of X that contains X and the empty set and is closed under intersection, union, and complementation forms a Boolean algebra. Stone’s representation theorem, proved in the 1930s, tells us that every Boolean algebra arises in this way:

Every Boolean algebra is isomorphic to a family of subsets of some set X that contains X and the empty set and is closed under intersection, union, and complementation.

For example, the proposition “it is the weekend” can be identified with the subset {Saturday, Sunday} of the set of days of the week. In general, each proposition can be identified with the set of possible worlds in which it is true. Logical operations then become set-theoretic operations.

In short, Stone’s representation theorem lets us view any Boolean algebra, however abstract, concretely as a family of subsets. This is a familiar perspective: we can use Euler–Venn diagrams, for instance. Moreover, if a Boolean algebra with 2n elements is too large to picture and describe directly, we can instead use the fact that it is isomorphic to the power set of a set with only n elements.

To turn this representation theorem into a genuine duality, we need two further ingredients.

First, among all possible representations of a Boolean algebra, one is canonical. Stone identified the properties that characterise it: it is the unique representation that is both separating and compact. These properties were already familiar to mathematicians because they belong to topology, the study of spaces. Stone thereby defined a particular class of topological spaces, now called Stone spaces. These are sets X equipped with a topology that identifies a separating and compact Boolean algebra of subsets of X. The correspondence between Stone spaces and Boolean algebras is the first ingredient of Stone duality.

The second ingredient concerns functions. Take two Boolean algebras and their corresponding Stone spaces. The significant functions between the algebras are those that preserve their Boolean structure: homomorphisms. Between Stone spaces, the significant functions are the continuous ones, a topological notion. Here is the crucial point: homomorphisms from the first Boolean algebra to the second correspond one-to-one with continuous functions from the second Stone space to the first. The functions correspond, but their direction is reversed.

This, then, is Stone duality: a correspondence between Boolean algebras and Stone spaces that also encompasses their functions, with the arrows pointing in opposite directions. It is an example of a categorical duality. In mathematics, a category consists of a collection of structures—such as Boolean algebras—and a chosen notion of functions between them—such as homomorphisms. A categorical duality is a mirror that places two categories in correspondence while reversing the functions.

This correspondence allows us to transfer a problem from one category to the other, choosing whichever setting makes the reasoning easier.

Pulling many dualities out of a hat

Stone duality is not an isolated case. Throughout mathematics, algebraic structures—sets equipped with operations—and geometric structures—sets equipped with distinguished subsets or relations—often face one another across a mirror. Algebraic geometry carries this connection in its very name. In logic, dualities resembling Stone duality are used to study a variety of non-classical logics.

During the decade following Stone’s result, functional analysts discovered dualities for several classes of algebraic structures resembling vector spaces equipped with a metric. These dualities show that such a metric vector space can be viewed as a set of continuous real-valued functions on a certain topological space. What kind of space? A compact Hausdorff space: a generalisation of a closed interval of the real line. Such spaces are a more “continuous” counterpart to Stone spaces, which are much more disconnected.

A particularly intriguing version of these dualities for compact Hausdorff spaces has recently been completed. To place it in context, we need to recall mathematicians’ interest in classes of algebras axiomatised by equations, that is, conditions of the form

∀ x1, …, xn   s(x1, …, xn) = t(x1, …, xn).

Here s and t are expressions that combine the variables with the available algebraic operations. Boolean algebras, for example, are axiomatised by equations. This guarantees a range of useful properties: they are closed under products, subalgebras, and quotients, and they admit free algebras, among other things.

As early as 1969, John Duskin had shown that the category of compact Hausdorff spaces is dual to some category of algebras axiomatised by equations. In a 2018 paper in Advances in Mathematics, Vincenzo Marra and Luca Reggio finally solved the open problem of finding an explicit axiomatisation of these algebras, known as δ-algebras, by exhibiting a finite list of equations that defines them.

Two shadows under one umbrella

We have encountered two kinds of duality:

  • Stone duality;
  • dualities for compact Hausdorff spaces, involving algebras that resemble vector spaces.

The final stage of our journey is the search for a common generalisation of the two.

Every Stone space is a particular compact Hausdorff space and therefore has an associated δ-algebra. This algebra, however, is quite different from the Boolean algebra associated with the very same space. It is as though we were using two very different mirrors and obtaining two dissimilar algebras from one space.

Yet these algebras do have a common generalisation. Vector spaces, which appear in several dualities for compact Hausdorff spaces, have an addition operation. In a Boolean algebra, the “union” x ∨ y behaves somewhat like addition, but only when x and y are disjoint. Why not introduce a genuine addition defined everywhere?

Doing so places a Boolean algebra inside a larger structure. For example, the Boolean algebra {0, 1} can be embedded into the integers ℤ, where the sum of any two elements is defined. More generally, every Boolean algebra can be embedded into an abelian group equipped with a lattice order, meaning that binary suprema and infima exist, and in which addition is defined everywhere. These ordered groups can then be specialised in either of the directions that interest us:

  • if the group is highly divisible—if fractions can be taken, as with the rational numbers—it resembles a vector space;
  • if the group is not very divisible, as with the integers, it resembles a Boolean algebra.

These ordered groups therefore generalise both Boolean algebras and the algebras occurring in dualities for compact Hausdorff spaces. A natural question follows: is there a duality for these ordered groups that generalises both Stone duality and the dualities for compact Hausdorff spaces?

The answer is yes. In a 2025 paper in Advances in Mathematics, Vincenzo Marra, Luca Spada, and I obtained a duality for the category of metrically complete unital abelian lattice-ordered groups. The idea is that such an ordered group can be represented as a particular subset of the continuous real-valued functions on a compact Hausdorff space X.

To determine which subset of functions to take, it is enough to equip X with a function into the natural numbers. This denominator function records, at each point, the “degree of divisibility of the distinguished element 1”.

For example, a space consisting of a single point can be assigned different denominators. With denominator 2, its corresponding ordered group is 1⁄2ℤ. With denominator 1, the group is ℤ, which is intimately related to the Boolean algebra {0, 1}. With denominator 0, the group is ℝ.

We thus obtain a duality between metrically complete unital abelian lattice-ordered groups and a-normal spaces—short for “arithmetically normal spaces”. These are compact Hausdorff spaces equipped with a function into the natural numbers that satisfies certain properties.

Within this encompassing duality, Stone duality is recovered by restricting to a-normal spaces in which every point has denominator 1. The duality for compact Hausdorff spaces is recovered by choosing denominator 0. The remaining cases correspond to ordered groups with intermediate characteristics.

One last reflection

We have crossed a small universe of mirrors: logic and sets, vector spaces and topological spaces, ordered groups and spaces with denominators. These are surprising dualities between distant worlds that appear to have nothing in common, yet contain the same information.

And this is precisely where the wonder begins. As John Baez and James Dolan wrote, an equation is interesting to the extent that its two sides are different. The identity sin2(x) + cos2(x) = 1 is far more intriguing than 1 = 1, precisely because its two sides have such different forms.