When Is One Thing Equal To Another? (via LtU)

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 superobject; 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.”

For the “object-oriented” programmer this perhaps speaks to the familiar concepts of abstraction and encapsulation: “objects” in the OOP sense are always super-objects in the sense of the above, and the question of the equality of two such objects can become rather vexed (see, e.g., Bloch’s Essential Java on when and how to override equals). So this passage holds some clue about the attractions of category theory to computer scientists.

I’m also interested in what this says about Badiou’s math?me. Are sets in some sense primitive, “the objects that we truly want”, or are they also super-objects, explicit presentations of entities that then appear only as “equivalence classes” between presentations?

Even before I describe category more formally, it pays to examine the category of sets as an example. The category of sets, though, is not just “an” example, it is the proto-type example; it is as much an example of a category as Odette is un amour de Swann.

[…]

In observing how mathematicians tend to use the notion class, it has occurred to me that this notion seems really never be put into play without some background version of set theory understood already. In short by a class, we mean a collection of objects, with some restrictions on which subcollections we, as mathematicians, can deem sets and thereby operate on with the resources of our set theory. I’m perfectly confident that that this seeming circularity can be-and probably has been-ironed out. But there it is.

[…]

Category theory doesn’t legislate which set theory we are to use, nor does it even give ground-rules for what “a” set theory should be. As I have already hinted, one of the beautiful aspects of category theory is that it is up to you, the category-theory-user to supply “a” set theory, a model bare category of sets S, for example. A category is a B.Y.O.S.T. party, i.e., you bring your own set theory to it. Or, you can adopt an even more curious stance: you can view S as something of a free variable, and consequently, end up by making no specific choice!

All very intriguing.