Some excerpts from Barry Mazur’s paper on When is one thing equal to some other thing?. This is a rough draft of the paper from January 2006, so there may be some awkwardnesses. Also, many sections from the original paper are left out, and the text below will not make sense as an article by itself. All the footnotes and other citations are omitted.

In particular, the whole discussion of replacing an object with its network of relationships (and the following discussion of object and representation) is interesting. It’s also probably worth following up on the bit about a Wittgenstenian interpretation of Yoneda’s Lemma. This paragraph from near the end sums up some of this well:

It sometimes happens that the introduction of a term in a mathematical discussion is the signal that an important shift of viewpoint is taking place, or is about to take place. An emphasis on “representability” of functors in a branch of mathematics suggests an ever so slight, but ever so important, shift. The lights are dimmed on mathematical objects and beamed rather on the corresponding functors; that is, on the networks of relationships entailed by the objects. The functor has center stage, the object that it represents appears almost as an afterthought. The lights are dimmed on on equality of mathematical objects as well, and focussed, rather, on canonical isomorphisms, and equivalence.

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The awkwardness of equality

One can’t do mathematics for more than ten minutes without grappling, in some way or other, with the slippery notion of equality. Slippery, because the way in which objects are presented to us hardly ever, perhaps never, immediately tells us — without further commentary — when two of them are to be considered equal. We even see this, for example, if we try to define real numbers as decimals, and then have to mention aliases like 20 = 19.999…, a fact not unknown to the merchants who price their items $19.99.

The heart and soul of much mathematics consists of the fact that the “same” object can be presented to us in different ways. Even if we are faced with the simple-seeming task of “giving” a large number, there is no way of doing this without also, at the same time, “giving” a hefty amount of extra structure that comes as a result of the way we pin down — or the way we present — our large number. If we write our number as 1729 we are, sotto voce, offering a preferred way of “computing it” (add one thousand to seven hundreds to two tens to nine). If we present it as 1 + 12^3 we are recommending another mode of computation, and if we pin it down – as Ramanujuan did – as the first number expressible as a sum of two cubes in two different ways, we are being less specific about how to compute our number, but have underscored a characterizing property of it within a subtle diophantine arena.

The issue of “presentation” sometimes comes up as a small pedagogical hurdle – no more than a pebble in the road, perhaps, but it is there – when one teaches young people the idea of congruence mod N. How should we think of 1, 2, 3, … mod 691? Are these ciphers just members of a new number system that happens to have similar notation as some of our integers? Are we to think of them as equivalence classes of integers, where the equivalence relation is congruence mod 691? Or are we happy to deal with them as the good old integers, but subjected to that equivalence relation? The eventual answer, of course, is: all three ways — having the flexibility to adjust our viewpoint to the needs of the moment is the key. But that may be too stiff a dose of flexibility to impose on our students all at once.

To define the mathematical objects we intend to study, we often — perhaps always — first make it understood, more often implicitly than explicitly, how we intend these objects to be presented to us, thereby delineating a kind of super- object; that is, a species of mathematical objects garnished with a repertoire of modes of presentation. Only once this is done do we try to erase the scaffolding of the presentation, to say when two of these super-objects—possibly presented to us in wildly different ways— are to be considered equal. In this oblique way, the objects that we truly want enter the scene only defined as equivalence classes of explicitly presented objects. That is, as specifically presented objects with the specific presentation ignored, in the spirit of “ham and eggs, but hold the ham.”

This issue has been with us, of course, forever: the general question of abstraction, as separating what we want from what we are presented with. It is neatly packaged in the Greek verb aphairein, as interpreted by Aristotle1 in the later books of the Metaphysics to mean simply separation: if it is whiteness we want to think about, we must somehow separate it from white horse, white house, white hose, and all the other white things that it invariably must come along with, in order for us to experience it at all.

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One of the templates of modern mathematics, category theory, offers its own formulation of equivalence as opposed to equality; the spirit of category theory allows us to be content to determine a mathematical object, as one says in the language of that theory, up to canonical isomorphism. The categorical viewpoint is, however, more than merely “content” with the inevitability that any particular mathematical object tends to come to us along with the contingent scaffolding of the specific way in which it is presented to us, but has this inevitability built in to its very vocabulary, and in an elegant way, makes profound use of this. It will allow itself the further flexibility of viewing any mathematical object “as” a representation of the theory in which the object is contained to the proto-theory of modern mathematics, namely, to set theory.

Defining Natural Numbers

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For only in terms of [the structure intrinsic to the whole gamut of natural numbers] (packaged, perhaps, as a version of Peano’s axioms) do we have a criterion to determine when your understanding of “natural numbers,” and mine, admit “faithful translations” one to another. A consequence of such an approach — which is the standard modus operandi of mathematics ever since Hilbert — is that any single mathematical object, say the number 5, is understood primarily in terms of the structural relationship it bears to the other natural numbers. Mathematical objects are determined by – and understood by — the network of relationships they enjoy with all the other objects of their species.

Objects versus Structure

Mathematics thrives on going to extremes whenever it can. Since the “compromise” we sketched above has “mathematical objects determined by the network of relationships they enjoy with all the other objects of their species,” perhaps we can go to extremes within this compromise, by taking the following further step. subjugate the role of the mathematical object to the role of its network of relationships — or, a further extreme — simply replace the mathematical object by this network. This may seem like an impossible balancing act. But one of the elegant – and surprising – accomplishments of category theory is that it performs this act, and does it with ease.

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Equality versus Isomorphism

The major concept that replaces equality in the context of categories is isomorphism. An isomorphism f : A → B between two objects A, B of the category C is a morphism in the category C that can be “undone,” in the sense that there is another morphism g : B → A playing the role of the inverse of f ; that is, the composition gf : A → A is the identity morphism [on A] and the composition fg : B → B is the identity morphism [on B] . The essential lesson taught by the categorical viewpoint is that it is usually either quixotic, or irrelevant, to ask if a certain object X in a category C is equal to an object Y . The query that is usually pertinent is to ask for a specific isomorphism from X to Y.

Note the insistence, though, on a specific isomorphism; although it may be useful to be merely assured of the existence of isomorphisms between X and Y , we are often in a much better position if we can pinpoint a specific isomorphism f : X → Y characterized by an explicitly formulated property, or list of properties. In some contexts, of course, we simply have to make do without being able to pin- point a specific isomorphism. If, for example, I manage to construct an algebraic closure of the finite field [with two elements], and am told that someone halfway around the world has also constructed such an algebraic closure, I know that there exists an isomorphism between the two algebraic closures but – without any further knowledge – I have no way of pinpointing a specific isomorphism. In contrast, desipte my ignorance of the manner in which my colleague at the opposite end of the world went about constructing her algebraic closure, I can, with utter confidence, put my finger on a specific isomorphism between the group of automorphisms of my algebraic closure and the group of automorphisms of the other algebraic closure. The fact that the algebraic closures are not yoked together by a specified isomorphism is the source of some theoretical complications at times, while the fact that their automorphism groups are seen to be isomorphic via a cleanly specified isomorphism is the source of great theoretical clarity, and some profound number theory.

A uniquely specified isomorphism from some object X to an object Y characterized by a list of explicitly formulated properties – this list being sometimes, the truth be told, only implicitly understood – is usually dubbed a “canonical isomorphism.” The “canonicality” here depends, of course, on the list. It is this brand of equivalence, then, that in category theory replaces equality: we wish to determine objects, as people say, “up to canonical isomorphism.”

Representing one theory in another

If categories package entire mathematical theories, it is natural to imagine that we might find the shadow of one mathematical theory (as packaged by a category C) in another mathematical theory (as packaged by a category D). We might do this by establishing a “mapping” of the entire category C to the category D. Such a “mapping” should, of course, send basic features (i.e., objects, morphisms) of C to corresponding features of the category D, and moreover, it must relate the composition law of morphisms in C to the corresponding law for morphisms of D; we call such a “mapping” a functor from C to D.

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In this way, we have a vocabulary for establishing bridges between whole disciplines of mathematics; we have a way of representing grand aspects of, say, topology in algebra (or conversely) by establishing functors from the category of topological spaces to the category of groups (or conversely): construct the pertinent functors from the one category to the other! The easiest thing to do, at least in mathematics, is to forget, and the forgetting process offers us some elementary functors, such as the functor from topological spaces to sets that passes from a topological space to its underlying set, thereby forgetting its topology. The more profound bridges between fields of mathematics are achieved by more interesting constructions. But there is a ubiquitous type of functor, as easy to construct as one can imagine, and yet extraordinarily revealing. Given any object X in any category C we will construct an important functor from C to the category of sets upon which C was built. This functor alone will be enough to “reconstruct” X , but – as you might guess – only “up to canonical isomorphism.”

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The object as a functor from the theory-in-which-it-lives to set theory

Given an object X of a category C , we can define a mapping (a functor that we will denote FX) that encodes the essence of the object X. The functor FX will, in fact, determine X up to canonical isomorphism. This functor FX maps the category C to the category S of sets (the same category of sets on which C is “modelled,” as we’ve described above). Here is how it is defined. The functor FX assigns to any object Y of C the set of morphisms from X to Y ; that is, FX(Y) := Mor(X, Y).

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The fundamental, but miraculously easy to establish, fact is that the object X is entirely retrievable (however, only up to canonical isomorphism, of course) from knowledge of this functor FX. This fact, a consequence of a result known as Yoneda’s Lemma, can be expressed this way:

Theorem: Let X, X’ be objects in a category C . Suppose we are given an isomorphism of their associated functors n : FX ~= FX’. Then there is a unique isomorphism of the objects themselves, h : X ~= X that gives rise – as in the process described above – to this isomorphism of functors.

The beauty of this result is that it has the following decidedly structuralist, or Wittgensteinian language-game, interpretation:

an object X of a category C is determined (always, only up to canonical isomorphism, the recurrent theme of this article!) by the network of relationships that the object X has with all the other objects in C.

Yoneda’s lemma, in its fuller expression, tells us that the set of morphisms (of the category C) from an object X to an object Y is naturally in one-one correspondence with the set of morphisms of the functor FY to the functor FX. In brief, we have (or rather, Yoneda has) reconstructed the category C, objects and morphisms alike, purely in terms of functors to sets; i.e., in terms of networks of relationships that deal with the entire category at once.

Representable Functors

The following definition (especially as it pervades the mathematical work of Alexander Grothendieck) marked the beginning of a significantly new viewpoint in our subject.

A functor F : C → S, from a category C to the category of sets S on which it is modelled, is said to be represented by an object X of C if an isomorphism of functors F ~= FX is given. The functor F is said, simply, to be representable if it can be represented by some object X.

If you consult the theorem quoted at the end of the last section you see that Yoneda’s lemma, then, guarantees that if a functor F is representable, then F determines the object X that represents it up to unique isomorphism.

One of the noteworthy lessons coming from subjects such as algebraic geometry is that often, when it is important for a theory to make a construction of a particular object that performs an important function, we have a ready description of the functor F that it would represent, if it exists. Often, indeed, the basic utility of the object X that represents this functor F comes exactly from that: that X represents the functor! Although a specific construction of X may tell us more about the particularities of X, there is no guarantee that all the added information a construction provides – or any of it – furthers our insight beyond guaranteeing representability of F.

Some of the important turning points in the history of mathematics can be thought of as moments when we achieve a fuller understanding of what it means for one “thing” to represent another “thing.” The issue of representation is already implicit in the act of counting, as when we say that these two mathematical units “represent” those two cows. Leibniz dreamed of a scheme for a universal language that would reduce ideas “to a kind of alphabet of human thought” and the ciphers in his universal language would be manipulable representations of ideas.

Kant reserved the term representation (Vorstellung) for quite a different role. Here is the astonishing way in which this concept makes its first appearance in the Critique of Pure Reason:

There are only two possible ways in which synthetic representations and their objects … can meet one another. Either the object (Gegenstand) alone must make the representation possible, or the representation alone must make the object possible.

It is this either-or, this dance between object and representation, that animates lots of what follows in Kant’s Critique of Pure Reason. With meanings quite remote from Kant’s, the same two terms, object and representation, each provide grounding for the other, in our present discussion.

Nowadays, whole subjects of mathematics are seen as represented in other subjects, the “represented” subject thereby becoming a powerful tool for the study of the “representing” subject, and vice versa.

It sometimes happens that the introduction of a term in a mathematical discussion is the signal that an important shift of viewpoint is taking place, or is about to take place. An emphasis on “representability” of functors in a branch of mathematics suggests an ever so slight, but ever so important, shift. The lights are dimmed on mathematical objects and beamed rather on the corresponding functors; that is, on the networks of relationships entailed by the objects. The functor has center stage, the object that it represents appears almost as an afterthought. The lights are dimmed on on equality of mathematical objects as well, and focussed, rather, on canonical isomorphisms, and equivalence.

[...]

Object and equality

A stark alternative – the viewpoint of categories – is precisely to dim the lights where standard mathematical foundations shines them brightest. Instead of focussing on the question of modes of justification, and instead of making any explicit choice of set theory, the genius of categories is to provide a vocabulary that keeps these issues at bay. It is a vocabulary that can say nothing whatsoever about proofs, and that works with any – even the barest – choice of a set theoretic language, and that captures the essential template nature of the mathematical concepts it studies, showing these concepts to be – indeed – separable from modes of justification, and from the substrate of ever-problematic set theory. Separable but not forever separated, effecting the kind of aphairesis that Aristotle might have wanted, for, as we have said, you must bring your own set theory, and your own mode of proof, to this party. With the other lights low, the mathematical concepts shine out in this new beam, as pinned down by the web of relations they have with all the other objects of their species. What has receded are set theoretic language and logical apparatus. What is now fully incorporated, center stage under bright lights, is the curious class of objects of the category, a template for the various manners in which a mathematical object of interest might be presented to us. The basic touchstone is that, in appropriate deference to the manifold ways an object can be presented to us, objects need only be given up to unique isomorphism, this being an enlightened view of what it means for one thing to be equal to some other thing.