Inside the Logic of “Products” and Equality in Set Theory

Table Of Links

Abstract

1. Acknowledgements & Introduction

2. Universal properties

3. Products in practice

4. Universal properties in algebraic geometry

5. The problem with Grothendieck’s use of equality.

6. More on “canonical” maps

7. Canonical isomorphisms in more advanced mathematics

8. Summary And References

Universal Properties

I am aware of three reasonable classes of foundations for mathematics: those based on set theory, those based on type theory, and those based on category theory. There are also various results saying that, broadly speaking, these foundations are able to prove the same theorems. However, for a discussion of equality to take place we will have to pin down exactly what we are talking about, and so I will choose ZFC set theory and classical logic as the basis for our discussion.1

The basis for this decision is simply that if a mathematician ever went to a class on the logical foundations of their subject (and many of them did not), then it was likely to have been a set theory class. Moreover, the pure mathematics literature is written in a superficially set-theoretic style: we are told that group is a set with some structure and axioms, a manifold is a set with a different structure and axioms, and so on. Here the word “set” is just a placeholder for the idea of a “collection of atoms”.

It is obvious that two sets which are equal have the same elements; this follows from the so-called principle of substitution for equality, which states that if X and Y are any two mathematical objects which are equal, then any claim which you can make in your foundational system about X is also true for Y . The converse, that two sets with the same elements are equal, is imposed as an axiom of the theory. This ensures that the abstract concept of a set coincides with our mental model of what it is representing: a set is no more and no less than a collection of stuff.

Now I would like to begin the discussion of properties which uniquely characterise a mathematical object. Let us start with an example: the product X × Y of two sets X and Y . Let me warn the reader now that in the next few paragraphs I will be making a very careful distinction between the concept of the product of X and Y , and the concept of a product of X and Y . The product X ×Y of two sets is defined to be the set of ordered pairs (x, y) with x ∈ X and y ∈ Y (one can check using the axioms of set theory that it is possible to create this set).

Note that here we run into the same issue that we saw earlier with the real numbers: there are several distinct ways to define the concept of an ordered pair in set theory. Set theory is designed very well to work with unordered pairs: the sets {x, y} and {y, x} have the same elements and are thus equal, so to define an ordered pair one needs to use some kind of hack.

The Wikipedia page for ordered pairs [Wik04b] currently gives three distinct constructions, due to Wiener ({{{x}, ∅}, {{y}}}), Hausdorff ({{x, 1}, {y, 2}}) and Kurotowski ({{x}, {x, y}}); all have the air of being slightly contrived.2 Again mathematicians are well aware that this issue does not matter at all in practice: all that we need to know is the defining property of ordered pairs, which is that (x1, y1) = (x2, y2) ⇐⇒ x1 = x2 and y1 = y2; this is all that we shall need, and all the definitions satisfy this property.

The product X × Y is equipped with two projection maps π1 : X × Y → X and π2 : X × Y → Y . Stricly speaking, it is not just the product, but the triple (X × Y, π1, π2) which satisfies the following universal property:

The universal property of products: A triple (P, π1 : P → X, π2 : P → Y ) is called a product of X and Y if it satisfies the following property: if S is any set at all, and f : S → X and g : S → Y are functions, then there is a unique function from S to P such that its composition with π1 is f and its composition with π2 is g.

The universal property is not a definition of the product of two sets; it can be thought of as infinitely many facts which a product needs to satisfy (one for each choice of set S and functions f and g). It is not hard to verify that the product X × Y of X and Y , equipped with the natural projections, is a product. But the converse is not at all true: there are typically plenty of other triples (P, π1, π2) which satisfy the property of being a product without being the product.

For example, if X = {37} and Y = {42} then the product X × Y is {(37, 42)}, but in fact any set P with one element, equipped with π1 : P → X sending everything to 37 and π2 : P → Y sending everything to 42, satisfies the universal property of being a product. In particular, there are in general uncountably many different things which satisfy the property of being a product. However mathematicians are extremely good at identifying these different things; they are “the same” in a manner which transcends the correct usage of the = symbol. We can do because of the uniqueness yoga for universal objects. Let us go through this yoga, which is a piece of formal category-theoretic nonsense, in the case of products.

Say P1 and P2 are both a product for X and Y . Applying the existence part of the universal property for P2 (with its projections to X and Y ) to the set S = P1 (with its projections to X and Y ) gives us a function α : P1 → P2 commuting with the projections to X and Y . Switching the 1s and 2s in the argument we can also construct a function β : P2 → P1 commuting with the projections. Moreover, β ◦ α is a map from P1 to P1 commuting with the projections, as is the identity function; by the uniqueness part of the universal property of P1 applied to P1 we deduce that β ◦ α must be the identity map on P1; similarly, α ◦ β must be the identity map on P2.

Hence α and β are bijections commuting with the projection maps. Finally, using the uniqueness part of the universal property for P2 applied to P1 tells us that α is the unique map from P1 to P2 which commutes with the projections, by symmetry β is the unique map from P2 to P1 which commutes with the projections. The upshot of this abstract nonsense (which never mentioned elements of any sets, just objects and morphisms) is that there are unique mutually inverse bijections between P1 and P2 which commute with the projections to the factors X and

Y . In particular, if P is a product of X and Y then it is uniquely isomorphic to the product X × Y of X and Y in a way compatible with the projections. In our one element example X = {37} and Y = {42}, if P is any one element set then the unique map from P to X × Y of course sends the element to (37, 42).

:::info Author: KEVIN BUZZARD

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:::info This paper is available on arxiv under CC BY 4.0 DEED license.

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