What is the space of all pairs of points? Which is its shape? Answering these apparently nonsensical questions will lead us into the language of modern algebraic topology

Apart from the empty set, a point is the simplest geometrical object you can imagine; however add another point and you run into some complex questions. Indeed, we have now introduced the possibility of interchanging the two points, and switching the two points of a pair is a reversible transformation: it can be undone by switching the points again. So we have not just added a point: we have moved from an object with only trivial self-transformations to an object with nontrivial self-transformations.

And this changes everything. So much that considering the space of all pairs of points naturally leads to the categorical language used by Galatius-Madsen-Tillmann-Weiss in their work on the homotopy type of cobordism categories, and by Hopkins-Lurie in their classification of topological field theories, two masterworks of contemporary algebraic topology (the branch of mathematics which uses algebraic tools to investigate the geometry of spaces). Simplifying a lot, these two results together describe the shape of the space representing the totality of smooth geometrical objects (e.g, circles, spheres, doughnuts) of every possible dimension, and classify its linear representations (this is akin to taking a photograph: the object we take a picture of is three-dimensional, and its photograph is only two-dimensional, so we loose a lot of information; but we can still read in the photograph many features of the original object).

The idea of a space representing all possible smooth objects is so abstract that even thinking of it may seem insane, not to mention wondering about its shape. But we can learn a lot on how to imagine and describe such a space by considering a toy model of it: namely, the space representing all possible pairs of points.

The first thing we have to learn about this space is that we should not think of it as a bare set of points: it is just too big for that. And too big things in mathematics are a formidable source of logical paradoxes (a celebrated one is Bertrand Russell's, which at the beginning of XXth century mined the foundations of naive set theory).

The impossibility of thinking of the space of all pairs of points in terms of bare sets is at first sight an overwhelming problem. Luckily, we live in an era which sees Alexander Grothendieck's prophecy come true: category theory is taking the place of set theory at foundations of mathematics. And, for the example we are concerned with, the categorical point of view simplifies things a lot. It says: “the pairs of points are only apparently infinitely many; actually, there is only one pair of points, reflected into indefinitely many replicas by indefinitely many mirrors; and there's a mirror reflecting a pair of points into itself”. Therefore, getting rid of replicas, we are reduced to a single object, “divided by two”. What is crucial here, is that we have completely forgot the nature of our objects, we are only remembering that we have a single object with exactly one nontrivial self-transformation. So, for instance, from a categorical point of view the space of all pair of points is also the space of all line segments of a fixed length: each such segment can be identified with another one, and reversing the segment is its only nontrivial self-transformation.

We have not said anything concerning the shape of this space, yet. To investigate this aspect of the problem, notice how we can say a lot of the geometry of a complicated space by looking at how simpler spaces can sit inside it. As an illustrative example, one can detect a doughnut's hole by noticing that on the doughnut’s surface a curve going around the hole can not be moved into a curve which does not go around the hole without breaking it. By contrast, any two closed curves on a sphere can be deformed one into the other, since a sphere has no holes. As we have remarked, the space we are investigating can be seen as the space of all segments of fixed length, so that a curve in this space can be thought of as a strip of paper. We can close a strip in two different ways which can not be deformed one into the other. Namely, we can just make a cylinder or we can make the strip have half a twist before closing it, producing the one sided surface known as Möbius strip. Therefore in our space there must be at least one hole. Refining this kind of analysis, one achieves a complete description of the geometry of our space.

Pairs of points have thus taught us an important lesson: to investigate any space representing a class of objects (a mathematician would say “a moduli space”) we only need to know which are its objects, and how any object can be related with itself (i.e, which are its self-transformations) and with the other objects. Once we know this, we know the shape of the space.