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#1




Is the equal sign overrated?
There's an interesting article in Wired. It's difficult to summarize. To me it seems silly on the surface, but I'm a mathematical lightweight.
https://www.wired.com/story/isthee...shashitout/ Quote:
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#2




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#3




That article was fascinating. It all sounds weird and arbitrary and confounding, but the process matches exactly much of what I've read about the history of math.
That's littered with major advances by geniuses which required acolytes who could act as gobetweens and make the abstruse accessible. It's always the next generation who grows up learning the new stuff without having to forget the old that builds of the new to advance the field into new territory. Schoolkids aren't going to need to worry about losing the equals sign. Equivalence is way beyond simple algebra. Most of modern math isn't even translatable into English terms that nonspecialists can follow. That's bad for mathematicians in one way, because a field is always deprecated when no one has any notion of what they do. But it's good for mathematicians in another way. They used to have to worry about cranks proving it was possible to trisect an angle or that 0.9999~ isn't equal to 1. But nobody is going to try to inflict their theories about Higher Topos Theory. They won't get past the title. 
#4




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Or to use a less abstract example, five pennies are equal to one nickel  unless you're replacing some blown fuses. Last edited by Little Nemo; 10142019 at 01:50 AM. 


#5




That sounds like a fiveyearold's logic: "Four is bigger than three, right? But what if the four is small and made of wood and the three is big and made of stone?"

#6




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It's really not that complicated (except when it starts getting highly technical). Consider, for example, a finitedimensional vector space, the same thing introduced in a firstsemester algebra course. Let's call it V. Next, consider the space of linear homomorphisms V → F, where F is the field of scalars. This is the "dual space" V*. Next, you can form V**. If you try to unwind the definitions, this double dual looks complicated and confusing, consisting of maps into a set of maps. However, V** is naturally/canonically isomorphic to V, and you may as well write V = V**, even though at first glance it might seem they are not the same. In other words, you can realize the same structure in several ways. Similarly when you study abstract algebra: now you have categories of all sorts of algebraic structures. But two different categories may be naturally equivalent to each other. (This might be really interesting, since the categories might arise in totally different ways.) More complicated algebraic structures might require the use of "higher category theory" as described in the article. 
#7




That's good to know
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#8




Yeah.
Sorry, DPRK but I'm afraid you didn't do a very good job of explaining this "really not that complicated" matter .
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#9




finitedimensional vector space: The one you are most familiar with is just the set of points in 3D space. How do you specify a specific point in your room? It has a height, a distance from one wall, and a distance from a rightangle wall. That's three coordinates (the finite part). The "space" is the set of all points that you can describe this way.
linear homomorphisms V → F: Suppose you're using your new coordinate system to describe the temperature in your room. For each point, you can use your thermometer to record a single value (a scalar). Suppose also we know that the temperature varies in a specific way: in particular, we know that the temperature can always be described as A*height+ B*wall1_dist + C*wall2_dist. Mathematicians call this a "linear" map. So we ask: what are all the different ways that we can map a point to a temperature (limiting ourselves to linear maps)? Or put another way, what are all the temperature distributions that our room can take on that can be called linear? Well, it turns out that the "space" of all these ways has the same number of dimensions as in the original vector space. That's not too hard to see here: the constants A, B, and C controlled the behavior of our map. There was one constant for each coordinate in the original vector. Slightly trickier: what happens when we do this operation twice? What is the space of all the different spaces that describe how to map a coordinate to a temperature? It's really hard to wrap your head around what that means, but it doesn't matter, because we know again that it will have three coordinates. Finding the "dual" didn't change the number of coordinates the first time, and it won't change the second or hundredth time either. But there's another aspect that's trickier to describe. I'll go with a simpler example: finding the dual space is a little like finding a number that, when multiplied by the original, equals 1. So, if we start with 3, then our dual number is 1/3, because 3*1/3=1. What's the dualdual of our number? It's just the original! 3 goes to 1/3, which goes to 3. When you go to higher dimensions, it's a little more complicated, but the basic principle is the same. Starting with some object E, we find a kind of inverse object E' such that E*E' = 1. And the same thing works in reverse, because E'*E = 1 too. That's what DPRK means then when he says that V = V^{**}, at least in some sense. The descriptions were totally different, one being a fairly commonsense thing and the other being a weird map of a map. And yet they're provably equivalent. (hope I didn't screw this up too much, so I welcome corrections) 


#10




Define "up".
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#11




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Actually, to give an example of two objects that look different but are, in an appropriate sense, "the same", does not take linear algebra (but I thought an algebraic example was appropriate since this is all heading in the direction of abstract algebra). For instance, in the world of sets you need to somehow define what are ordered pairs, like (a, b). One way to do that is to define (a, b) as { {a}, {a,b} }. But you could have just as well defined it as { {b}, {a,b} }. It looks (and is) different, but is the same thing as far as anyone is concerned. In fact, in abstract algebra, one is concerned not merely with "objects" (e.g., sets), but rather with categories of objects (sets plus functions between sets). So you have the category of finite sets, the category of all sets, the category of finitedimensional vector spaces, the category of groups, and so on. Back to Dr. Strangelove's point: if you have two objects in a category, say A and B, sometimes they are isomorphic in the sense that there is a map from A to B, call it f, and also a map g from B back to A, such that f and g are inverses of each other, so gf = fg = 1. In the case of sets, this simply means that A and B have the same number of elements. But if you have two different categories, say C and D, then we want to say when they are the same this is called "equivalence" so you will have F: C → D and G: D → C, except now it is not true in general that G(F(x)) = x and F(G(y)) = y, instead, for any x, G(F(x)) will, in general, only be "naturally isomorphic" to x in the sense I started describing for ordered pairs and for double duals of finitedimensional vector spaces. The problem now is, you may need to consider categories with additional structure, which amounts to categories of categories, and check whether they are equivalent, and so on to higher levels. Even to write down the set of diagrams and equations such an equivalence must satisfy quickly becomes unwieldy (we are talking something like 50 pages of diagrams just to define a 4category) and then impossible, which is why the kind of topologyinspired technical machinery introduced by Lurie and others is indispensable. 
#12




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That sort of thing becomes important once you consider the "category" of all (say, finitedimensional) vector spaces, or, more abstractly, any category, and you want to define what it means to be equivalent to some other category. Technically, this means that you have maps (functor) F: C → D and G: D → C, but now FG and GF are not the identity, they are only naturally equivalent to the respective identity functors. SUMMARY VERSION: if you are doing abstract algebra, your objects are not concrete and you can't really say when two things are "equal", only that they are equivalent in an appropriate sense, therefore the equality sign gets a bit abused. 
#13




Just use some new symbols. We had to do that in OO programming decades ago. The equal sign is fine, don't blame the symbol if you are using it the wrong way.

#14




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You may be using terms that aren't in commonly understood by non mathematicians. If it is impossible to explain in simpler terms then that's fine, but I'm not learning much from your explanations as currently stated. In fact, looking back at the original extract quoted by davidm I think I got the concept pretty well from that and I'm not understanding what you are saying well enough to know whether you are even for or against it!
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#15




→
You have "objects", like x, and you have "maps", like x ^{f}→ y. These satisfy various axioms; for example, if f: x → y and g: y → z, then you can compose them to get a new map gf: x → z, and this composition is associative, so h(gf) = (hg)f. (This is for now really an equals sign, since maps between two objects are supposed to be elements of a set, and in set theory there is the notion of whether two elements are the same or not.)
In linear algebra, vector spaces are the "objects" and linear maps are the "maps", aka homomorphisms. The thing is, with just this level of category theory, it is easy to keep track of all these axioms and the complexity requiring "higher category theory" does not yet emerge. You do not need it to do basic linear algebra. We do already see some abuse of notation, as it were, when people write things like V = V**, though. In fact, from the linear algebra point of view, considering the "dual" of a vector space and the "tensor product" of two vector spaces does bump you one notch up the "higher category theory" ladder when you try to enumerate all the different coherence conditions these now have to satisfy. Just a little, though. In general, a higherlevel category will not just have objects and maps between objects, but also maps between maps, and maps between maps between maps... it's not something one would seem to encounter studying linear algebra, but this sort of thing, but where you have to go to infinity, does seem to arise naturally in the category of topological spaces, homotopy theory and situations like that. Or maybe I am misinterpreting the thrust of the article, I don't know... 
#16




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Apparently, what the article is talking about is something new; but whether it will catch on and turn out to be important, I cannot say. 
#17




But in some cases it's true. Suppose you have a small quantity of paint on your brush and you're painting a number. You might have just enough to paint a seven. But you might run out if you try to paint a six. Six takes more paint than seven even though it's a smaller number.

#18




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It's like how classical geometry was based on the assumption that everything was happening on a twodimensional plane. It worked just fine. But in the nineteenth century, some mathematicians began exploring how geometry would work on a different type of surface. 
#19




Thudlow Boink's example of == vs === is right on, illustrating the difference between equality and equivalence.
But (feel free to correct me if I'm wrong!) I don't think this is a matter of classical versus nonclassical, more like moving from the rigid world of sets, where two sets are either equal or not equal, to richer structures, where this issue comes up. This Lurie stuff is related to algebraic topology, if I understand correctly, where spaces or functions may not be exactly the same but may be deformable into each other. Higher category theory also comes up in physics as topological quantum field theory, also naturally in abstract algebra as soon as you consider tensor products (at least as a baby example). Anyway it seems to crop up in various contexts, so it's good to know it when you see it. 


#20




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#21




I don't get it. From about juniorsenior 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 welldefined 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 (oneone 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)
https://arxiv.org/pdf/math/0608040.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 theorybutnotatthepointit'stopologyyet). 
#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. Last edited by Jragon; 10142019 at 08:10 PM. 


#25




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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 semiubiquitous so it's not out of the question that some of this technical machinery will also find use outside the realm of algebraic topology. Quote:

#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.
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#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 wellequipped 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 tongueincheek 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.
Last edited by Jragon; 10172019 at 07:15 PM. 


#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 crosssection 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 2category. 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 3category, 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_{∞}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 distinctlooking categories into a single one, is there any suggestion that the number of interesting categories is also finite and/or open to classification? 
#32




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#33




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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 finitedimensional simple Lie algebras, etc.
So there should be setups where you want to classify certain 2categories, for example instead of just algebras you look at various categories of modules. And probably on to higher ncategories... 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.
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#36




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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. Last edited by DPRK; 10182019 at 09:06 AM. 
#37




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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




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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.
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#40




One good illustration/application of higher categories/not forgetting about isomorphisms could be the 1950s1960s 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 nontrivial symmetries. The solution is to classify curves as points of a "moduli stack" (which remembers these symmetries) rather than a plain moduli space.

#41




Here is another example of equal not quite working 1/21/3+1/41/5+1/61/7... = X
1/2+1/41/3+1/6+1/81/5 +1/10+1/121/7+...=Y they both eventually add 1/even and subtract 1/odd but X is not equal to Y. 
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