Is the equal sign overrated?

Great Debates
21

I don’t get it. From about junior-senior undergraduate courses on, equivalence relations replace equality anyway, or did back when I was in school at various points between 25 and 45 years ago. One set, or algebraic group or ring, or open set, would be equivalent to another with respect to some property. E.g. isomorphism is the term for two sets that are equivalent in terms of their properties with respect to abstract algebra. But nobody claims that isomorphic means equal.

Equal was reserved for two things, really: (1) sets etc. that not only were equivalent, but had the exact same elements, so there could be no property - algebraic, analytical, topological, whatever - that they were different with respect to; and (2) finite numbers.

But finite numbers themselves don’t play a big role in higher math, with some exceptions like combinatorics and number theory, because they’re pretty simple things. 4=4, nobody’s going to argue with you. That doesn’t mean that four apples = four oranges, the ‘=’ just means you have the same number of apples as you do of oranges. That’s all they’re telling you.

22

The first sentence of the quote already gets my back up. He talks about two quantities being equal and then segues into two sets being equal. Equality of sets is well-defined and means they have exactly the same elements. But two distinct sets can, of course, have the same number of elements. We might write |X| = |Y| or choose some other convenient notation but we do not say X = Y. I think Cantor used a macron (overline) for it. Now if you want to say that there are six distinct isomorphisms (one-one correspondences) between two sets of three elements, go right ahead. Mathematicians have been doing that for centuries (maybe millenniums). But if he thinks this is a marvelous new discovery that revolutionizes mathematics, he’s got another think coming.

I really fail to see what category theory has to do with all this.

23

Here’s the actual book Higher Topo Theory on arxiv (PDF)

For anyone that wants to make more sense of the intro instead of relying on a Wired article. I can’t easily make sense of it myself without the background in category theory (my higher math is all complexity theory and graph theory-but-not-at-the-point-it’s-topology-yet).

24

If my limited ability to interpret this is right, what it’s getting at is far more subtle and technical that any reduction to lay understanding can communicate. He’s actually saying that there are, in fact, multiple classes and levels of granularity where “=” really, truly applies. For instance X =/= Y directly, but there is a sense where they’re in two different classes that are provably equivalent by any reasonable rigorous definition. And then this is stackable, X =/= Z, and Y=/=Z, and the class X or Y are in =/= the class Z is in, but the equivalence class containing X and Y is equal, in some rigorous provable sense, to the class Z is in.

You can phrase examples in lay understanding but it necessarily makes it seem kind of arbitrary and “well duh” without the rigorous backing in hierarchies of category equivalence. In one sense “2+2” is equal to “two plus two” but in another sense it isn’t, for instance. == vs === is a decent way to put it, but it’s farther reaching than that.

I think it’s not worth looking at this like some revolution against the concept of strict equality, but rather a tool for better taxonomizing and classifying equivalence relations in higher order contexts far, far beyond the idea of basic sets and numbers BUT once you apply it that far, it becomes clear that it has some ramifications as far as basic algebra/set theory that haven’t been considered.

25

The sign ≅ seems to be used in some algebraic contexts. Maybe not in set theory, where people are more likely to talk about the cardinality of various sets.

I’m pretty sure the business about the equal sign was deliberately meant to be a catchy title.

Not sure specifically why he (the Quanta guy) thinks Lurie’s books-- which the article is about— “revolutionize mathematics”, but it is true that even higher categories are semi-ubiquitous so it’s not out of the question that some of this technical machinery will also find use outside the realm of algebraic topology.

The books certainly make central the notion of an “infinity-category”.

26

As mentioned computer science is a good example.

A shortint and a longint may both = 8, so they’re equivalent on that level, but if they’re both the same type, for example shortints, then you have a higher level of equivalence. If they’re different names for exactly the same object then it’s an even higher level of equivalence. Then there’s inheritance, where they can be equivalent in some aspects.

27

I downloaded the book linked by Jragon and read the intro. First, there is no word there about = being obsolete. It is, in fact, a book about higher order topos (not topo) theory by a highly regarded young mathematician. Second, the purpose is to understand aspects of cohomology theory (of which I know something. And even the intro demonstrated to me how much the world of category theory has passed me by.

28

I’ve been meaning to weigh in here but haven’t had the time until now. First my bona fides: I’m an algebraic topologist, follower of and occasional participant in a mailing list that includes most or all of those quoted in the article (in fact, a link to the article was posted on the mailing list gleefully pointing that out), and one of those quoted was my thesis advisor lo these many years ago, and a collaborator on several projects. But I’m also one of those who hasn’t had time to really learn the machinery behind infinity categories, though I suspect they would help with some projects I’m currently attacking with more primitive tools.

There’s a lot to unpack to try to describe how a general audience article relates to the actual mathematics being done. As usual, there are inaccuracies and wrongly placed emphases, but I’ve seen a lot worse.

In this post, let me talk about the issue of equality: This really has to do with category theory itself, which goes back to the 1950s, not infinity categories in particular. A category (as originally defined) is a collection of “objects” with a collection of “arrows” connecting some of them. In the prototypical examples, the objects are sets with some additional structure, and the arrows are functions that preserve that additional structure. Like functions, arrows can be composed to give other arrows and composition is associative; each object also has a distinguished identity arrow that behaves as the identity for composition. Now two objects are essentially the same if there are arrows connecting them in both directions such that both composites are the corresponding identity arrows. We say that the two objects are isomorphic. For all intents and purposes, the two objects will behave exactly the same within the category. With caveats.

This all becomes more interesting and important when you look at functions from one category to another, taking objects to objects and arrows to arrows. Just like other mathematical gadgets, we’re only interested in those functions that preserve the structure we have, namely identity arrows and composition. The jargon is that functions that preserve that structure are called functors, and this was part of the original reason for looking at categories: There were constructions floating around that, say, took topological objects and produced algebraic ones, and it turned out that these constructions were functors from a category of topological spaces to one of algebraic things, and pointing that out clarified and simplified what people had been doing. (Not that mathematicians adopted category theory eagerly when it was first developed; it was referred to somewhat snidely as “general abstract nonsense,” a moniker the practitioners adopted with some pride.)

But there were also questions like this: Suppose that F and G are two functors. When would we say that they always give the same result? For each object X on which they’re defined, we get new objects F(X) and G(X). Now, they may be constructed in rather different ways and it usually doesn’t make much sense to ask if F(X) = G(X), i.e., that they give exactly the same object in the target category. The more useful question to ask is if F(X) and G(X) are always isomorphic objects. Even then, there may be many possible isomorphisms between each F(X) and G(X), so it’s even more useful to know that there’s a canonical way of picking isomorphisms that are compatible in a reasonable sense when we vary X. This gives us the notion of natural isomorphism between two functors, and in a very real sense, category theory was invented to give a language in which we could define and use natural isomorphisms.

Bringing this back to equality: The upshot is that, within a category, it is almost always an uninteresting question to ask whether two objects are equal, and in a sense it’s a question that category theory isn’t well-equipped to answer and doesn’t really care about. The interesting question is whether two objects are isomorphic and, because there are usually multiple possible isomorphisms, to either determine all the isomorphisms or pick out one that’s particularly interesting for some reason (like being part of a natural isomorphism). Trying to ignore the choice of isomorphism and just declaring isomorphic objects to be equal almost guarantees misery somewhere down the line. The tongue-in-cheek adage is that equality is evil. (There are objections from a number of category theorists to the use of this language.)

29

Oooooh, natural transformation is something I actually (somewhat) understand. That makes a lot more sense now, thanks.

30

Now let me say a little about infinity categories. The Quanta article makes it sound like these sprang unexpected and fully grown from the brow of Jacob Lurie, but his work is more a culmination of developments that had been going on for quite a while already.

The article makes a decent attempt to explain an issue that algebraic topologists had been dealing with for a long time, in its discussion of paths on a sphere. One thing we like to do is look at a topological space (for example, a sphere) and two points on that space, and then look at paths from one point to the other. If there is a path, that’s good, but how many paths are there? Well, we really want to say that two paths are the same (to us) if we can deform one path into the other continuously, keeping the endpoints fixed. We say that the two paths are homotopic, and we’re really interested in how many paths are there up to homotopy, i.e., after we identify paths that are homotopic. Consider a space more complicated than a sphere, like the surface of a donut. Given two points, there are many different ways to connect them by a path, as you can wrap around the small circular cross-section any number of times, in either direction, or around the big circle of the donut any number of times, in either direction, or some combination of those, before getting to the other point. It turns out that the number of windings around each of the two circles determines whether two paths are homotopic or not.

Looking at the points in a space as objects and homotopy classes of paths as arrows, we get a category, called the fundamental groupoid of the space (the name is, of course, jargon, and not particularly pretty). Composition of arrows here amounts to following one path and then following another, considering the result to be a path from the very first point to the last.

But sometimes this is too crude. We’ve ignored the fact that, even if two paths are homotopic, they may be homotopic in different ways. We can ask whether the deformations we use (which we call homotopies) are themselves homotopic. And the answer may well be that there are several distinct ways in which two paths can be homotopic. So perhaps we should build a gadget that remembers not just paths between points, but also homotopies between paths. One way to do this is to take the fundamental groupoid and replace the set of paths between each pair of points with a category, in which the paths are the objects and arrows between paths are (homotopy classes of) homotopies between paths. This changes the fundamental groupoid from a category into what’s called a 2-category.

But this also forgets some of the information that’s present in the space, because, even if two homotopies between two paths are themselves homotopic, there may be several distinct such homotopies available. Adding in that information gives a 3-category, and so on, to infinity categories.

Well, “and so on” turned out to be overly optimistic. To actually do this correctly and define carefully what an infinity category really is turned out not to be at all easy or obvious. Category theorists and algebraic topologists had been working on various proposals for how to do it since at least the last decade of the 20th century. If I understand correctly, what Lurie did was settle on one particular way to do it and then develop in exquisite detail how that model works, and that it does work to do what people had been wanting.

The example of the “fundamental infinity groupoid” sketched above is just one application. There are quite a few other places where we knew that we needed some way to deal with homotopies, homotopies between homotopies, homotopies between those, and so on all at once. Various devices were developed, one common one involving what are called E[sub]∞[/sub]-operads, which works but is somewhat cumbersome. If we can come to an agreement about how to define infinity categories, and work up expositions that are readable and usable (arguably, Lurie’s are not that, yet), they will subsume and replace a lot of earlier approaches and simplify and clarify a lot of earlier work. That’s the promise and the hope.

31

Please tell me if this question makes no sense:

It seems like category theory is a kind of more general form of group theory. The classification of the finite simple groups was a major effort and basically showed that there are only a finite number of interesting groups.

Given that there’s a new tool available that can demonstrate equivalence between categories, and might collapse wide ranges of distinct-looking categories into a single one, is there any suggestion that the number of interesting categories is also finite and/or open to classification?

32

I think the answer is, far from it. The categories of interest are often very large, in the sense that the collection of objects is a proper class, too large to be considered a set. For example, the category of all sets: The collection of all sets can’t be itself a set, without running into Russell’s paradox. No time at the moment to get into this further, but it’s a very different classification problem than something as “limited” as the classification of finite groups.

33

Ok, thanks. To be clear, I didn’t mean to imply that there would be a direct analogy between the two. Groups of course come in infinite varieties, but they’re “boring”: they’re parameterized, and when you understand one you understand them all. And then there are the non-simple groups.

I get that the objects in a category are “big”, but maybe that implies some alternate parameterization, like through the surreal numbers?

34

Just like finite simple groups, there are all sorts of algebraic classification problems like finite-dimensional simple Lie algebras, etc.

So there should be setups where you want to classify certain 2-categories, for example instead of just algebras you look at various categories of modules. And probably on to higher n-categories… but you are not trying to classify “all categories” or “all groups” or “all symmetric monoidal categories”, it seems like you need some restrictive condition before you can expect something like a simple enumeration or parameterization, without it being wild or uncomputable or similarly unanswerable in a satisfactory way.

35

Well, it’s great stuff, but it is totally irrelevant to 99.999999% of the people in the world who get along just fine with Euclidean Geometry and basic mathematics.

36

What makes you trust these 99+% polloi types to have a competent grasp of Euclidean geometry and other basic mathematics?

Again, it’s just a catchy title —no one is suggesting there is literally a problem with the equals sign as taught to schoolchildren or that they need to bone up on homological algebra to get through secondary school.

37

So? I can find a million articles online that would be irrelevant and uninteresting to 99.999999% of the people in the world. Heck, I’ve written many of them. But the 70 people for whom they are relevant and interesting fall all over them.

Do you read every single thread that has ever been posted on the Dope? If not, do you go into all the threads you don’t read and complain they aren’t relevant to you? Why not try it and see what reception you get.

38

To put it another way: You can ask the classification question for just about any class of objects you’re interested in. But the ones you can actually answer tend to be those that are fairly tightly constrained. For example, asking to classify all groups, without any cardinality restriction, would be hopeless. Even the classification of all finite groups is really only partially done with the classification of the finite simple groups.

The more constraints on your objects, or, somewhat paradoxically, the more complicated the objects are, the better your chances of being able to classify them.

39

I started this thread and it’s gone way way over my head. That’s not a complaint, just an observation.

I am humbled by the members of this forum.

40

One good illustration/application of higher categories/not forgetting about isomorphisms could be the 1950s-1960s geometrical idea of considering “stacks” instead of ordinary spaces. The difference is that instead of simply having points, as in if you solve the equation x² + y² = 1 you get a set of points, you keep track of isomorphisms and have a groupoid instead. For instance, if you fold a space, as in the orbifolds that show up in physics (e.g., take a piece of paper and fold it in two, or form it into a cone), you naturally get a “quotient stack”. You can even algebraically construct the classifying stack of a group, by letting it act (trivially) on a single point and taking the quotient, because the stack remembers the extra data even though it looks like you still have a single point. This was also used to great effect in algebraic geometry, for example to classify elliptic curves. You can begin by parameterizing them by complex numbers, but when you form them together into a space you soon run into problems due to the existence of curves with non-trivial symmetries. The solution is to classify curves as points of a “moduli stack” (which remembers these symmetries) rather than a plain moduli space.