“A way that math can make the world a better place is by making it a more interesting place to be a conscious being.” So says mathematician Emily Riehl near the start of this episode, and it’s a good summary of what’s to come. Emily works in realms of topology and category theory that are far away from practical applications, or even to some non-practical areas of theoretical physics. But they help us think about what is possible and how everything fits together, and what’s more interesting than that? We talk about what topology is, the specific example of homotopy — how things deform into other things — and how thinking about that leads us into groups, rings, groupoids, and ultimately to category theory, the most abstract of them all.
Support Mindscape on Patreon.
Emily Riehl received a Ph.D in mathematics from the University of Chicago. She is currently an associate professor of mathematics at Johns Hopkins University. Among her honors are the JHU President’s Frontier Award and the Joan & Joseph Birman Research Prize. She is author of Categorical Homotopy Theory, and co-author of the upcoming Elements of ∞-Category Theory. She competed on the United States women’s national Australian rules football team, where she served as vice-captain.
[accordion clicktoclose=”true”][accordion-item tag=”p” state=closed title=”Click to Show Episode Transcript”]Click above to close.
0:00:00.0 Sean Carroll: Hello everyone, welcome to the Mindscape podcast. I’m your host, Sean Carroll. We’ve done a lot of sort of science on the podcast over the years, as it were. The podcast is old enough I can now say over the years. Well, that’s pretty impressive. And doing science, when you try to talk about science to an audience which is not necessarily full of specialists in that particular kind of science, what do you do? How do you talk about science to non-scientists? Well, the very first thing you do is you strip out all the math, right? You assume that the people listening are not necessarily familiar with the equations, the symbols, the manipulations that the professional scientists have to go through to understand their field. But you can nevertheless, even without the math, you can talk about the concepts that the scientists are dealing with. This raises a puzzle when the thing you wanna talk about is math. And we haven’t talked about math as a pure subject that much here on Mindscape. We did talk to Steven Strogatz who’s a professional mathematician, but he’s very close to being a physicist in many ways. We haven’t quite ascended to the truly abstract realms of very pure mathematics.
0:01:05.9 SC: So, that’s what we’re doing here today. Pure mathematics. No equations, but we’re gonna try to talk about the concepts that you come across at the most elevated realms of mathematical thought. Our guest today is Emily Riehl, who is a topologist at Johns Hopkins, and she’s a very big believer in being able to conceptualize these deep mathematical ideas in the simplest and best way possible, so she’s a great person to guide us on this tour. And the foundational idea we’re gonna be talking about is topology. Topology is the study of the properties of mathematical spaces that are invariant when you smoosh them, when you smoothly deform them a little bit. Like if you have some clay that is molded into a shape, you can talk about the number of holes in the shape of clay that you have, and then if you move around the clay a little bit, if you don’t rip it, the number of holes will remain constant, so that’s a topological invariant. It turns out, and this is where the fun part comes, that as a mathematician, you wanna say, “Okay, what do we mean by characterizing the qualities of a mathematical space that are invariant under smooth deformations?” Well, you can count the holes, you can also say, “Well, how many times can I make a path that wraps around the holes in this particular system?” Those kinds of structures turn out to be numbers or transformations or other things that we can add together and multiply together.
0:02:35.2 SC: And we build these mathematical algebraic structures by asking the question, “How do we topologically characterize these kinds of spaces?” So we’re lead from topology into algebra, and Emily’s gonna take us there. We’re gonna discuss things like homotopy, groups, rings, groupoids, all these words that you’re not supposed to have necessarily been familiar with already, we’re gonna discuss them. And by the end, we’re getting into what is called category theory. Category theory is something where even the other mathematicians go, “Oh no. That’s too abstract for me.” Category theory is sort of a general theory of spaces and structures and maps between them. It provides a different way of thinking about mathematics as a whole. So I’m a big believer that math, just like physics, is part of the general intellectual conversation we should be having. There should be history and economics and math and physics, and it will be a mistake to exclude math from this overall intellectual discourse. And I think this conversation is a good example of how we can do it. Thinking like a mathematician gives you new handles on the world, just like thinking like a physicist does, and so I think it’s gonna be fun. Let’s go.
[music]
0:04:04.5 SC: Emily Riehl, welcome to the Mindscape podcast.
0:04:06.7 Emily Riehl: Thank you.
0:04:07.9 SC: I know that as a physicist, as a theoretical physicist, who thinks about the universe and the many worlds of quantum mechanics and so forth, I’m often asked, “What is the point of this? Are you making a better cell phone or are you curing cancer? Why are you doing this?” And I have my own answers, but as a mathematician who works on topology, category, and things like that, you must also get this. What is your… Do you have a favorite answer to that kind of question?
0:04:33.2 ER: Sure, my dad loves to ask me what the practical applications are of all the math that I love to think about. And I think he knows that he’s kind of needling me because it’s just not the point for all mathematicians. I mean for some of course it is, there are very important uses of mathematics to make the world a better place, but I guess a way that mathematics can make the world a better place is by making it a more interesting place to be a conscious being, and that’s how I… Or that’s what inspires me to be a mathematician.
0:05:07.0 SC: Good. Though I think… I’m not gonna say that’s the right answer because like you say, different people have different answers, but that’s analogous to my answer. When people ask me about looking for the Higgs boson, and so many of my physics friends will say, “Well, it might some day inspire a new technological something.” I’m like, “No, no. Number one, no, it won’t. And number two, that’s not why we’re doing it.” We’re doing it ’cause we wanna find out what’s going on. And if there’s a application someday, that’ll be a benefit.
0:05:32.5 ER: Sure.
0:05:33.1 SC: The other preliminary question I wanted to ask… I just had a podcast episode a couple of weeks ago about the philosophy of mathematics, and there’s realism versus non-realism, Platonism versus, I don’t know, anti-Platonism. I’m told that most working mathematicians are Platonists. They think of what they study as in some sense real. Do you know or care or have a take on that debate?
0:06:00.0 ER: I certainly don’t know as much about it as some of your other guests, but I agree with that instinct. The things that I think about and I dream about and I talk about with my friends feel very real to me. And I don’t expect that I’ll kind of trip over one on my walk to campus, but they feel as real to me as me as anything else. I guess, as I understand it, that a table that seems like a solid real thing is not really, really there, so… So yes, yeah, I subscribe to Platonism.
0:06:32.6 SC: Well, yeah, there’s certainly some structure to math. We all agree, given the axioms, what, where we go and so forth, so there’s something there. I actually don’t have a strong opinion one way or the other. That’s why I’m quizzing people these days, but…
0:06:45.3 ER: Though actually, I mean, there’s a philosophy of mathematics that’s maybe a little closer to the point of view of category theory, which we’ll get into later on, which goes by the name of structuralism, which says that what a mathematical object really is, is determined by the role that it plays within mathematics. So for example, you could ask, what are the natural numbers? Zero, one, two, three, four, five. And a role that the natural numbers play within mathematics, I could give it a fancy name, I could say they’re the kind of universal discrete dynamical system, but essentially what that means is natural numbers are something you can recursively define functions on. If you’re trying to define a sequence of points or a sequence of real numbers, what you… Of strategy…
0:07:46.4 ER: Fibonacci sequence, for instance, is a famous sequence, 1, 1, 2, 3, 5, 8, 13, 21, 34, the relative terms converge to the golden ratio. There’s a lot of fun properties in Fibonacci, but a strategy for defining a sequence is you define the first term in the sequence or maybe the first few terms, and then you give a formula or a strategy for producing the next term in the sequence from the terms you have previously. And the fact that recursion is possible is telling us something very deep about the role that natural numbers play in mathematics. And so I guess I’m of the point of view of that’s what the natural numbers are. They are sort of the thing that you can recursively define functions on. How quite… How that fits with Platonism isn’t entirely clear to me, but…
0:08:35.5 SC: Yeah, I don’t know, me neither. I did have a guest a while ago, James Ladyman, who’s a philosopher who says the same thing about physics. He is a structural realist. He thinks that what really matters are the different structures, not the objects that we attribute these structures sort of relating between… I don’t know.
0:08:51.3 ER: Great, it sounds like he’s a category theorist, so…
[chuckle]
0:08:54.1 SC: Could be. Aren’t we all, are category theorists? We’re moving in that direction. So we will get to there, but I think that you had… As we were talking about what to discuss here, I think you had a strategy that makes perfect sense, of starting with topology, because topology is something everyone has heard of, right? Why don’t I, rather than trying to guess, why don’t you tell us what do you think the definition of topology is that we could all get our hands on?
0:09:18.5 ER: Great. So topology is the study of spaces, both physical, geometrical spaces that we move in in the world, but also spaces that you might imagine that have some resemblance to Euclidean space, three dimensions, two dimensions, one dimension, but are somehow more exotic. So a topological space could be, for instance, the points on a plane is an example of a topological space, or you can imagine the points that are on the surface of a sphere, so the surface of a soccer ball, the surface of the Earth, or you can imagine the points that are on the surface of a donut. You can imagine that you’re an ant that sort of walks around a donut and all of the different configurations that ant could be in, those describe points in a topological space, and then you can start to compare what those worlds are like. And topology is a realm to do that.
0:10:20.9 SC: Can I just ask very quickly, when mathematicians use the word space, obviously it doesn’t simply mean the three-dimensional space in which we live. What’s the difference between space and set?
0:10:30.8 ER: Right. So if I wanted to define a topological space formally in mathematics, I would start with a set of points. So if I wanted to describe three-dimensional Euclidean space, I would start with the set of points which are determined by three real coordinates, sort of a distance along the x-axis, a distance along the y-axis, a distance along the z-axis. So that’s a set of points, so we can think of them as the points in three-dimensional space. But then what turns a set into a topological space is also a way to understand distances between points, or more generally understand when points are nearby and when points are far apart. So that could be a function that tells you how to compute the distance between two points, that’s a metric space, which is an important type of topological space, or there’s a more abstract way to get at the same thing in the absence of a formula for distances.
0:11:28.0 SC: Okay, and so a line is a space, a circle is a space, the sphere… So for mathematicians the sphere is just the surface of the sphere, it’s not the interior. Like you said, the two-dimensional sphere is the surface of the ball. And then you could have a like 100-dimensional sphere or whatever. So there’s a lot of spaces we have to play with.
0:11:48.6 ER: You can have Mobius bands, so that’s a space that’s kind of like a cylinder, so it’s like a toilet paper tube, but imagine you cut the toilet paper tube sort of long ways from end to end, so now you have just a flat strip, and then you twist it and then glue it back together, so that is also a space. It’s quite similar to the cylinder, but it’s a non-orientable space. It’s just confusing in a Mobius strip, whether you’re on the inside or the outside, there’s no way to decide that. Whereas on the toilet paper tube, there’s the outside where the toilet paper goes around and then there’s the inside which you sort of put on the roll, on the holder so…
0:12:33.5 SC: Do you have one of those Klein bottle sculptures? I have a Klein bottle sculpture.
0:12:37.8 ER: I don’t, but…
[chuckle]
0:12:40.3 ER: Right. A Klein bottle is a one-dimension higher version of this, and what’s interesting about the Klein bottle is… Well, so it’s a surface that’s kind of like the surface of a sphere. Let me describe how you would build a Klein bottle.
0:12:58.3 SC: Sure.
0:13:00.9 ER: You can start with a flat sheet of paper. You would glue two sides of the paper together, so you’re forming now a toilet paper tube, and then… So from there, a simpler construction that isn’t the Klein bottle is you could just stretch the toilet paper tube and glue the end circles together, and if you did that you would get a surface of a donut shape. So that’s familiar.
0:13:25.1 SC: The torus. We can use the word torus. It’s okay. [laughter]
0:13:26.9 ER: Okay, we can use the word torus. And if I want to get a Klein bottle, it’s the same idea. I’m gonna glue the two ends of the toilet paper tube together, glue those two circular ends together, but I want to twist the one. So if I’m gluing… If I’m walking around… When I glue them together, I’m zipping up the… Going counterclockwise around one of the circles and clockwise around the other circle, but I wanna reverse the orientation on one of the circles before doing this.
0:13:52.8 SC: Right.
0:13:52.9 ER: Now, you can’t do this in three dimensional space. And so if we were in a fourth dimension, you could actually embed a Klein bottle into four dimensional space and do this construction, but there are these toys that you should just… Everyone should just Google.
0:14:07.1 SC: Yep.
0:14:07.3 ER: And they are these toy glass versions. And what’s fun about it is there is some ambiguity between what space is inside of a Klein bottle and what is outside. So if you have a torus, like a glass donut, you could fill it with jelly, and the jelly is inside the torus, it’s not outside. Until you bite it you’re not gonna get jelly everywhere.
0:14:27.0 SC: We all agree, right. Yeah.
0:14:28.3 ER: But if you have one of these Klein bottles, you could try and put jelly, or let’s say Kool-Aid inside the Klein bottle, and if you’re not careful, if you flip it upside down and flip it right side up again, it’s gonna get all over your chest, because there’s no inside and outside. Yeah.
0:14:45.1 SC: And this is the stock-in-trade of topologists, right? So you don’t care so much about the individual bumps and wiggles like geometry cares about, topology is more loosey-goosey in what it cares about.
0:14:56.4 ER: Exactly, yeah, very loosey-goosey. [chuckle]
0:15:00.3 SC: But the other thing to get on the table, I guess, is that there are topological spaces that we care about… Well, maybe they’re inspired by real physical situations, but you might think that well, physical space is three-dimensional, and we can have subsets of that, therefore, you know, there’s some topological spaces to think about, but not that many, but as you point out, like there are complicated situations where the space of all possible configurations of something is an interesting topological space.
0:15:26.5 ER: Yeah, totally. I mean, one of my favorite sources of examples of topological spaces are something called configuration spaces, and these come up in robotics, they come up in physics, where they might be called state spaces, and exactly as you pointed out, these can get very high dimension very quick, but let me mention an example that’s small enough that we can dream about it. Imagine you have a factory and there’s a one dimensional track on the floor in the factory. So there’s a strip, essentially, and maybe it’s kind of a… Or like a train track, and you can move forward and backwards along that track. And we’re gonna have two robots positioned on the track. There’s a red robot, and a blue robot. Well, let’s actually start with one robot. If you had one robot on a track, then you can describe its positions using just one dimension. It’s like an interval. You can have the robot all the way at the one end, or all the way at the other end, or any place in between. It can move continuously. So this one-dimensional interval is describing the space of configurations of that robot. So now, let’s put two robots on the track, a red one and a blue one, and now there’s a space that describes the configurations of those two robots. I mean, you might think it’s a one-dimensional space ’cause you’re still…
0:16:47.4 SC: Right.
0:16:47.8 ER: The robots are still on this one-dimensional track, but actually, you should use one coordinate to keep track of where the red robot is, and another coordinate to keep track of where the blue robot is, and so that is describing the space that’s formed by taking a product of an interval with itself, in other words, it’s a square, including the interior, except the two robots can’t be in the same spot. One… If these are physical robots…
0:17:13.3 SC: They take up space, yeah.
0:17:13.8 ER: They can’t collide into each other. So you have to remove a diagonal segment from that square, and what you’re left is with these two triangular components. The bottom triangle corresponds to the red robot being on the right of the blue robot, and the top triangle corresponds to the red robot being on the left of the blue robot. And so what’s interesting about the space, what we can see already by visualizing the configuration space, which is this square with the diagonal removed from it, is that it’s disconnected. Once you’ve put your robots on the track, you’re not going to be able to swap their orientation with respect to each other.
0:17:49.0 SC: That’s right.
0:17:50.0 ER: But if you’re designing the factory, you might say, “Well, that’s a little bit undesirable. I don’t have as much flexibility,” and improvement, perhaps, in the design depending on, of course, what you’re trying to do with these robots and this track, is you could make a circular track instead of a straight track. A circle is also a one-dimensional object, you just move around it going clockwise. There’s one direction of motion. But if you had two robots on a circular track, then the way you’d describe their configuration space is you would take each robot as some arbitrary point in the circle. We have two circles though, so we would take the product of those two circles and… This is fun to think about, that actually, the product of two circles corresponds to the points on the surface of a torus again.
0:18:35.8 SC: Right.
0:18:37.2 ER: So S1 cross S1. The product of two circles are the points on the torus. Again, the robots can’t be on the same point, so what you have to do is you have to imagine you have the surface of a torus, you built it by gluing together your toilet paper tube. Now you’re gonna make a diagonal cut that… It’s a cut that goes around, sort of both loops in the torus once, so you… I actually did this at home last night to test it in the way my toilet paper tube was made.
0:19:04.7 SC: You’re an experimentalist, good.
0:19:05.7 ER: Have the seam exactly, exactly there so it’s easy to see that cut edge, a sort of a diagonal cut that moves once around in both directions. And if you do that and you’re left with a connected surface, and that corresponds to the idea that if you have two robots on a circular track, they can really visit all configurations. There’s no sort of left and right anymore, you can have them in any position. So anyway, this is just sort of baby, baby examples of a really rich and beautiful subject.
0:19:32.3 SC: Right, because it’s good because what topologists care about are the features that don’t change by small deformations of the space. So the fact that a space is in two disconnected components like the original example versus just being in one connected component, like the second example, that’s the kind of thing that gets topologists very excited.
0:19:51.1 ER: Right, so when I say circular track, it could be an oval track, it could be a square track, it could be a hexagonal track. From a topologist, those are all the same.
0:19:58.9 SC: The famous thing I bet many people have heard that to a topologist, a coffee cup is the same as a donut, right? ‘Cause they’re both toruses. [chuckle]
0:20:06.1 ER: Right. That’s not my favorite example because… So topologists have different notions of when things are the same that we’re alluding to. And one of the notions of sameness is if two spaces are what’s called homeomorphic. And what homeomorphic means is that there is a sort of one-to-one correspondence between the points and the space. There’s a way… There are two invertible functions, invertible continuous functions that map all the points on one space to the other and the other to the one. And so if you have a solid torus now built out of clay and you have a solid coffee cup, you can define one of these homomorphisms, this sort of one-to-one correspondence between the points in the torus and the points in the mug that are continuous so that you’re not tearing the surfaces apart, or adding new holes where there weren’t holes before, that sort of thing.
0:21:06.0 ER: So it’s true that topologists consider a torus and a coffee mug to be the same, but for a really kind of obvious reason. They’re just so evidently the same, but there’s another notion of sameness, a weaker notion of sameness that also comes up in topology, which is called homotopy equivalence. And here the idea is two spaces are homotopy equivalent if you can continuously deform one to another. And so let me give you some examples. So all of Euclidean space, three-dimensional Euclidean space is homotopy equivalent to a single point, so you can imagine… I think of this homotopy equivalence is like the reverse Big Bang. So you imagine three-dimensional Euclidean space, so you have… Let’s imagine we have the origin point, so the center of the universe, and then all the other points in the universe. And I realize the universe is probably not three-dimensional, but let’s just bear with me here. [chuckle]
0:21:57.4 SC: Well also, it doesn’t have a center, but that’s okay, we’re gonna… You’re a mathematician…
0:22:00.7 ER: And [0:22:00.8] ____ does have a center, but let’s pretend there’s a center of the universe and the universe is three-dimensional. So in the reverse Big Bang, what I’m gonna do is I’m gonna continuously move all the points in the universe back in a straight line to the center of the universe, to the origin, and they’ll move faster or slower depending on how far are away they are. So at time zero, every point is where it is, and then at time one, they’ve all crashed home into the center of the universe. So that is describing a continuous deformation that reveals that three-dimensional Euclidean space and the point are the same. So here’s an example, sort of everyday example of a homotopy equivalence that feels more fun to me than the…
0:22:37.6 SC: Well, I’m sorry. Sorry, just to… Let me just butt in there because… So just to get it clear, the reason why this is an interesting example is because the point and three-dimensional space are homotopy equivalent, one can be continuously deformed to the other, but they’re not homeomorphic because there’s not sort of an equal number of points in them.
0:22:56.0 ER: Absolutely. Yeah, right, there’s uncountably, infinitely many points in three-dimensional space, but there’s one point in one-point space and… Yeah, so this is a cooler, way more destructive notion. I mean dimensions totally don’t matter anymore. All n-dimensional Euclidean space is also homotopy equivalent to a point. So an example, an everyday example that is kind of like that is, to a topologist, a pair of pants and a thong are the same.
[laughter]
0:23:23.0 ER: You can imagine a homotopy equivalence that sort of shrinks sort of vertically, I guess, and just leaves the thong from the pair of pants.
0:23:34.0 SC: And so the point is that there are different ways of expressing the idea of sameness depending on your sort of level of focus on what you care about.
0:23:42.1 ER: Yeah, totally, totally. And this is very pervasive in mathematics. Mathematicians love trying to understand the sense in which things are the same, and it’s often a pretty loose sense.
0:23:52.7 SC: So I had a very, very brief career, in fact, as a hack topologist when I was in graduate school, and I needed to calculate the homotopy of various things that appear in particle physics, because we have these things called topological defects, which are very important from the early universe. None of the things I calculated turned it out to be relevant to the real world, but that’s the risk that we take when we do these things, but it gets us into… So far it’s been fun, homotopy, smoothly deforming things into each other. But now we wanna be a little bit more rigorous about it. We wanna characterize, we wanna use this notion of smoothly deforming one thing into another to characterize the topology of different spaces. It’s in some sense our job to sort of characterize all topological spaces?
0:24:41.0 ER: Yeah, exactly. A classical problem, which is a good way to visualize this is you can ask how many different surfaces are there, sort of in this very flexible sense, so not necessarily just… So for instance, is… We’ve described a few different surfaces, there’s the sort of surface of the sphere, there’s a surface of a torus, you can have many-holed… The torus was the donut shape. You can imagine having many-holed donuts or like you’re baking a tray of donuts, but you’ve put them too close to each other, and so they’ve all kind of congealed, and there’s one donut with three holes and that’s another surface. And there’s a classification question, how can we tell that these spaces are really different? I mean they seem different, perhaps, maybe I’m giving away…
0:25:29.9 SC: Yeah, no they are. Yeah. [chuckle]
0:25:30.0 ER: My intuition, but they seem different. But how do we prove that they’re different because we’ve seen that these wildly different spaces are in some sense the same.
0:25:39.9 SC: Yeah, so how do we do that?
[chuckle]
0:25:43.1 ER: So yeah, so the idea is… This is a hard question to answer geometrically, because just because you can’t imagine a continuous deformation of one space to the other doesn’t mean it doesn’t exist. So a strategy is to bring algebra into the story somehow and give a way to assign a number or some other kind of algebraic structure to a topological space, and in a way that would give the same answer if the spaces were homotopy equivalent. So the homology groups or homotopy groups that… There are various different constructions you can do, and it’s easy to prove that if the spaces are continuously deformable into each other, they’ll produce the same invariance, and so if you happen to get different invariance than you know that they were different.
0:26:36.1 SC: Right, so actually…
0:26:37.1 ER: So let me…
0:26:38.1 SC: Go ahead.
0:26:40.0 ER: Well, I can give an example of this.
0:26:43.4 SC: Yes, exactly. Do that.
0:26:44.0 ER: So what I’m gonna try and argue for you is that the surface of a sphere, so the surface of a soccer ball is a different space than the surface of a donut. So a torus and a sphere are different topological spaces. And just to make sure we’re imagining these spaces, again, imagine you’re an ant walking around the surface of a sphere, that’s one space, the different positions the ant could be, or you can imagine an ant walking around the surface of a donut.
0:27:11.3 SC: Not the interior in either case, just the surface.
0:27:13.2 ER: Not the interior in either case, or the ant can’t eat the donut or… The ant is sort of stuck on the surface, and it can’t jump off either it’s… Okay, so the invariant that I’m gonna use to prove that these are different is something called the fundamental group that is… And what it’s counting essentially, so in each… For convenience, I’m gonna pick a point on each of these spaces. These spaces are connected, so picking a point is not a big deal, so pick whichever one you want. We’ll take the north pole on the sphere and any point at all on the torus. And what the fundamental group is going to calculate is the number of essentially different ways that the ant can walk from this home base point, walk around on the surface and come back to the home base point.
0:28:05.1 SC: Right.
0:28:06.2 ER: So the subtlety there is I said, the number of essentially different ways. So there are uncountably, infinitely many ways an ant can walk around these surfaces, but what I’m asking are they essentially different. And so let’s set aside the sphere example for now. Let’s start by thinking about the surface of the donut. So an example of two ways that are essentially different is we can imagine starting at our home base point and walking, let’s say in a counterclockwise direction around the loop that goes through the sort of donut hole, if you put a donut hole inside the donut that’s… You can imagine walking, it’s sort of in a vertical…
0:28:42.8 SC: The short way around, basically.
0:28:44.2 ER: The short way around, great. So that’s one way the ant could walk or the ant could walk, let’s say, tracing the same trajectory, but going in reverse orientation, going backwards rather than forwards. So those are fundamentally different, and what I mean by fundamentally different is there’s no way to continuously deform the first trajectory into the second trajectory while staying on the surface of the donut. If I could sort of slice through where the dough bit of the donut is, I could do that. I could sort of cut through the bit that’s not on the surface, but that’s not allowed. So walking on the surface once the ant decides it’s going clockwise rather than counterclockwise, those are different choices that the ant can make.
0:29:37.3 SC: In fact, they are homotopically inequivalent, right? Because you cannot continuously deform one into another.
0:29:40.8 ER: S2: Homotopically inequivalent. Right, and there are… What’s cool is you can actually enumerate all of the different choices. So another different trajectory the ant could take is it could walk the long way around. So let’s say staying on the top and going clockwise or going counterclockwise, or it can do some sort of combination where it loops some number of times through the center circle while also traversing the long way around. And there’s a way to enumerate all of these essentially different trajectories. What you need are two different integers, essentially. One integer that’s describing the number of times you go in a clockwise or counterclockwise direction around the short loop, and then another integer is describing the number of ways to go counterclockwise or clockwise depending on whether it’s positive or negative around the long loop, and you can prove that this pair of integers enumerates all possible trajectories.
0:30:39.2 SC: Right, right.
0:30:40.9 ER: So on the sphere, it’s actually quite a lot simpler, so if you have… You can imagine any path the ant might take on the surface of the sphere from the north pole to the north pole, kind of wandering around any which way, and you could kind of shrink that trajectory in a continuous way so that it gets smaller and smaller and smaller. So if it in fact goes all the way to the south pole, maybe it shrinks a little bit, so it only goes to the equator, and then maybe it shrinks a little more so it stays north of the tropic of Cancer and so on and so forth. And eventually, all of the trajectories are deformable to just the ant sitting at the north pole and never moving at all. So there, there’s only one trajectory that the ant can make, and that’s the proof somehow that these two spaces are not the same.
0:31:33.0 SC: Which is wonderful because it took us on a little journey. I will actually wanna dig into this because on the one hand, we’re not surprised that the sphere and the torus are topologically different, but it was a bit of an effort there to actually show it, even that case where we had things under control. And in doing that, you sample across… So you ask the question, how many different ways are there to sort of do this path that returns to itself, a loop, a closed circle in the space. And in the case of the torus, you just uncovered this pair of integers, [chuckle] the number of times you go around the short way, the number of times you go around the long way. And integers, not to get too fancy about it, now that’s algebra, that’s not geometry or topology or anything.
0:32:17.5 ER: No.
0:32:18.0 SC: Roughly speaking, mathematicians are either geometers and they like space, or they’re algebraists and they like equations, but an algebraic structure appeared here somehow.
0:32:30.1 ER: Right, I think this is just the tale of complexity in the modern world. Everybody has to be everything these days. So even if you really… Your heart isn’t with the geometry, you have to use some algebra.
0:32:42.0 SC: Well, I wanna just take advantage of your being here to dig into this a little bit more…
0:32:46.9 ER: Sure.
0:32:48.9 SC: We ask this question about how many ways you can draw a circle roughly speaking, in the torus. And there’s a lot of ways, but then there’s this hidden structure, in that, if you have a circle going around once in one direction and another circle just doing the same thing, you can add them together in some sense, and that’s the beginning of algebra. Am I too dramatic there?
0:33:12.4 ER: Yeah, that’s exactly right. So, Moore… I’ve described this… In describing the fundamental group, I said, “Let’s imagine we have a home base point, and we’re gonna consider the ant walking in these little loops from the home base to the home base,” but it’s actually maybe more natural to not have a home base point because I don’t know where we would be on the surface of a torus, for instance. So another way to think about this algebraic structure, you can imagine an ant walking from any point P on the surface, to any other point Q on the surface, it takes some path, and then later, maybe the ant is tired and takes an nap, and then later it wakes up and then it walks from the point Q to some other point R. And what you’re referring to, is you could then compose those two paths, compose those two trajectories, and then get a path directly from P to R. So if you can walk from P to Q and walk from Q to R, you can compose those two walks and get from P to R. And that’s the beginning…
0:34:11.6 ER: So this is a composition operation on the paths and the surface of a torus on any space, paths in any space. And really, the invariant that Moore exactly describes the algebra that’s hidden in the geometry is something called the “fundamental groupoid” or the “fundamental infinity-groupoid”, which is not a group, which is a type of mathematical objects we teach undergraduates, but an infinite dimensional analog of a group, an infinity-groupoid, which is a frontier of research-level mathematics.
0:34:48.5 SC: Right. I think that if… Look, if people are gonna listen to this entire podcast episode, I want them to come away knowing what an infinity-groupoid is. I think that’s gonna be something that they’ll be able to impress their friends with at cocktail parties and so forth. But to go very slowly to get there, you mentioned the fundamental group, which is the collection of the circles we started with. So what’s a group?
0:35:09.9 ER: Right. So a group is a… A way to… [chuckle] There’s this metaphor of the blind men and the elephant, when somebody’s holding… Somebody’s touching the trunk, somebody’s touching one of the legs, somebody’s touching the tail and have a completely different perspective of it, and that’s how I’m thinking about groups. So it’s really hard to know how to start. But a group in this context is a set. Here, it’s the collection of all different loops that an ant could take walking through some space together with composition operations. So it’s performing one loop and then following it by another loop, that can be understood as a loop even though you come back through the center, it’s still a loop, and then satisfying some very natural axioms. So if you’re walking along a loop, you could always reverse your trajectory and walk back in the other direction, and that’s an undoing somehow of the process. So every element in a group has an inverse that if you compose with it, it gets back to where you started, and a few simple axioms like that.
0:36:26.2 SC: So it’s kind of a, it’s a stripped down version of the integers or something like that, where the integers, you can add them together so the integers are an example of a group, right?
0:36:36.7 ER: Right. Integers with addition is an example of a group, absolutely.
0:36:38.0 SC: With addition. That’s right. Not with multiplication…
0:36:41.1 ER: Matrices with multiplication is an example of a group, matrices with addition is an example of a group. I have to say something about the dimensions to make those examples work, but…
0:36:52.3 SC: Okay. And the physicists love group theory because symmetries are a group, like rotations and translations and things like that.
0:37:01.0 ER: Yeah, absolutely. So another perspective on groups, if we move around the elephant, is a group is an axiomatization of the symmetries of an object or the different configurations that an object might be. So let me explain what a symmetry is. So imagine you have a twin mattress and you know that you’re supposed to flip your mattress occasionally ’cause I guess it’s good for the life of the mattress. And so you might wonder how many different ways are there to flip the mattress, how many different configurations could the mattress be in. And well, one option is whatever the mattress is when you started, then you could rotate it head-to-toe, so you’re switching the head and the toe but the top stays the same, that’s one move. You could also flip it side to side, so I’m keeping the head at the head and the toe at the toe, but I’m switching the top and the bottom up, or you could combine those two operations and the effect of this is the head is now where the toe was and the toe is where the head and the top and bottom surfaces are also flipped.
0:38:09.9 ER: So a group is recording on the one hand, the four different positions that the mattress could be in, but also how these different flipping operations that I described compose to each other. So each of those flips is in order two element of the group, meaning if you perform the same move twice, you get the mattress back to where you started. If you do any two different of the flips, you’ll get the third one, which is not a typical property of a group, but it’s special to this one, which goes by the name the “Klein four-group”. This group does not have a generator, meaning that there’s not one operation you can do over and over again that will take you all the way through the group. And this is why it’s hard to remember how to flip your mattress because you have to remember how you flipped it last month so you don’t just do the same operation again and get back to where you were the month before.
0:39:02.1 SC: So the integers do have a generator. You just add one and then you can get all the integers, either by doing that or undoing it, whereas…
0:39:07.0 ER: Exactly. Yeah, add one or negative one, you get all the way through everything. So yes, the integers are a cyclic group, which have a single generator, absolutely, but Klein four-group is not.
0:39:17.0 SC: Right. So the integers, maybe to people who are not group theory aficionados, are a nice little paradigm, but it’s important that groups can be very different. It’s a finite group, this mattress-flipping group, right? How many elements are there in the mattress-flipping group?
0:39:32.0 ER: Four.
0:39:32.4 SC: Four elements, okay, yeah, alright.
0:39:36.9 ER: These groups are super cool and really tell you something profound about geometry. So you might have heard of the platonic solids, which are the three-dimensional figures that you can get by gluing together regular two-dimensional figures. So a regular two-dimensional figure is like a triangle or a square or a pentagon or a hexagon where all of the sides have the same length, all the angles are the same, and so on and so forth, and there are infinitely many of these ’cause there’s a heptagon and an octagon and a nonagon for any natural number, n, you can get a plane figure with n-sides. So you might wonder how many of these regular platonic solids are, how many different shapes you can get by gluing together, say, triangles or gluing together squares, or gluing together pentagons, or gluing together hexagons maybe. And you can prove in fact that there are only finitely many, in fact, there are five of them using group theory by studying the symmetries, the orthogonal groups, the special orthogonal groups that describe the different configurations of these hypothetical shapes. Before even knowing they exist, you can limit the possible configurations.
0:40:52.4 SC: But clearly that’s not what Plato did.
[chuckle]
0:40:56.7 ER: I don’t know, maybe somewhere buried in Plato’s intuition. But another fun fact is… So the five… There’s something called the tetrahedron, which is built from triangles, and then there’s the cube, which is… That’s the most familiar one, built from squares, then there’s the octahedron, also built from triangles, and then the dodecahedron and the icosahedron. And even though I named five things, in a sense, there’s only three of them because there’s this duality relationship between the platonic solids which you might have seen. If you take a cube, so it’s got four… Sorry, its faces are squares, there are six of them, and they’re glued together along 12 different edges and there are eight corners. And what I’m gonna do is I’m gonna build a new platonic solid by replacing each of the corners by a face, and each face by a corner, and what you get in that just by connecting everything up is an octahedron, so one of the other platonic solids. And so there’s a duality relationship between the cube and the octahedron and also between the dodecahedron and the icosahedron and between the tetrahedron and itself, and those are reflected by their symmetry groups. So because of this duality relationship, the symmetry group that describes the configurations of the cube is the same isomorphic to… Same shape as the symmetry group of the octahedron and so on.
0:42:21.3 SC: Anyone who played Dungeons & Dragons knows about the platonic solids ’cause they have a dice, but they don’t know about the duality relations, so now, that’s good, that’s something else that they’ll have in their bag now. I’m only gonna do this at great risk, but I figured like, “Alright, while we’re talking about groups, why don’t we also explain rings and fields to the audience since these are the other algebraic structures that mathematicians love to throw around?”
0:42:45.1 ER: Yeah, absolutely. What’s funny is, after a while, you forget that these terms refer to other things. So when I say “field”, I definitely think about the mathematical field before I remember that’s also the thing that’s out the window, but right. So a group is describing a setting where you have a collection of objects and you have one composition operation to combine them together. So if we have the integers, we can think about addition, if you add two integers, you get another integer. But there are other binary operations on the integers that come up, multiplication for instance, and a ring is a setting where you have two operations, an addition operation or an addition-like operation, and a multiplication-like operation, and they interact in ways that are familiar for the integers. So if I…
0:43:39.5 ER: You can understand, there’s a distributivity property that says if I add two integers together and then I multiply by something, that’s the same as multiplying first and then adding, there are a few axioms like that. So what’s cool about… You might ask like, “Why do we bother? Everybody knows what the integers are, why do I need this abstract concept of a ring?” And that’s a totally fair question, but a really fun… An interesting fact is, there is a very deep analogy between the integers and polynomials. So a polynomial is… This is the sort of thing that you would meet in a high school algebra class. So you have a variable x, an indeterminate variable x, and then you can form a polynomial by adding up x’s and multiplying x’s and then throwing in real numbers as the coefficients. So the polynomial might be like 5x minus 5x plus, I don’t know, 17x squared plus pi-x cubed ’cause we can have real numbers as coefficients, minus 3.
[chuckle]
0:44:50.3 ER: So that’s a polynomial in a single variable x, and polynomials also form a ring. If you have two polynomials, you can add them together, you can multiply them, there’s rules for doing this that you might have learned in a high school algebra class, and the ring of polynomials with coefficients in a field, which we’ll get to what a field is later on, coefficients and the real numbers and the ring of integers are quite similar as rings, they’re both… They have a division algorithm, you can do long division with polynomials like with rings, all the ideal elements are principles. And so that’s the thing that’s of interest to mathematicians are these deep analogies between structures that are superficially quite different, yeah.
0:45:36.2 SC: And what’s the difference between a ring and a field ’cause they’re kind of similar?
0:45:39.5 ER: Right. So a field is… Like a ring, it has two binary operations. But if we go back to the ring of integers, there’s a pretty big difference between the addition rule and the multiplication rule, in that, every addition, every element has an additive inverse. So if I pick my favorite integer, 17, there’s another integer, -17, that when I add them together I get zero, which is the identity for the addition operation. But that doesn’t work for multiplication, if I pick my favorite integer, again, 17, there’s no integer I can multiply 17 by to get back to one, which is the multiplicative identity.
0:46:23.0 SC: Right.
0:46:23.9 ER: So that’s what distinguishes a ring, where you don’t necessarily have multiplicative inverses from a field. In a field you do have an inverse to both the multiplication and the addition operation, assuming you’re not trying to divide by zero. That never quite works. So things like the rational numbers, which throw in these multiplicative inverses, or the real numbers are fields in addition to being rings.
0:46:45.9 SC: And does the existence of the number zero get you in trouble because it doesn’t have an inverse?
0:46:52.2 ER: Yeah. In a field you have to treat zero as a special case.
0:46:57.0 SC: Okay.
0:46:57.1 ER: So in a field, you have to… Your zero and your one can’t be the same otherwise the whole thing collapses. And zero will not have a multiplicative inverse, but every other element has to.
0:47:08.1 SC: Got it, okay. And so when we started, we started this little journey, this whole side track, because we were thinking about the torus and its topology, and we found that the space of all the little loops the ant could walk on formed a group. Are there examples where we associate rings, or fields with topological invariance?
0:47:30.9 ER: That’s a good question. I don’t know that that’s commonly done, and I think the reason is that groups are just so rich, you know? [chuckle]
0:47:37.5 SC: Yeah.
0:47:39.0 ER: But a single group will not capture the full data of a topological space. I mean, really, what you have to do is introduce lots of different groups that are measuring lots of different things. So the fundamental group tells you about loops in the space, and whether you can… If you walk along a circular path in the space, can you fill that in with a disc? Could you… I guess, I don’t know how to say it any better than that. [chuckle]
0:48:08.6 SC: I see what you’re saying, yeah, can you contract it?
0:48:11.4 ER: Yeah, if you have a wire ring, is there a… Could you put a soap bubble on it somehow so that the surface of the soap bubble lives within the space that you were talking about? That’s not possible if you’re walking the short way around on a torus, because the soap bubble would have to cut through the dough, and that’s not on the surface, but it is on the surface of a sphere, you could just paint over the disc. So that’s measuring this one-dimensional holes, I guess, is the area that’s bounded by a one-dimensional sphere, but as we mentioned, there are spheres in higher dimensions. So you could ask, if you have a balloon inside your surface, can that be filled in with sand, staying within the surface? And that’s a two-dimensional analog of the same question, and there’s a question like that in all positive integer dimensions, and that family of groups describes the full homotopy type of a space. A single group does not, but if you have this infinitely many groups, it does.
0:49:14.8 SC: One of the things that I say just very casually to people who are non-mathematicians is that you might think that mathematicians spend all their time thinking about mathematical objects like spheres, or toruses, or whatever, but really, they’re thinking about maps between the different objects. So I guess it’s fair to say that first thing, the fundamental group is maps from circles into the space you care about…
0:49:37.6 ER: That’s right.
0:49:38.3 SC: Then there’s the set of maps from spheres, and then the set of maps from three spheres, and it clearly generalizes to infinite numbers.
0:49:47.1 ER: Yeah, absolutely, absolutely.
0:49:48.9 SC: But there’s something special about the groups. You don’t get fields, or rings which have two different binary operations on them. I guess the last thing that I have in that angle is, is there something special about fields and rings that have two binary operations that it’s worth stopping there? Can we define three binary operations on a set and make hyperrings or something like that?
0:50:12.0 ER: Sure. I mean, algebra is a very flexible subject, and to define what an algebra is in full generality you need the collection of elements that you’re considering, and then you can specify arbitrarily many operations, with arbitrary errorities and you can specify arbitrarily many rules between them. And the subject of universal algebra invites you to consider examples like that.
0:50:35.0 SC: Let’s invite listeners in the audience who are inclined in that direction to follow and study universal algebras, but we’re gonna go back to the topology, and the groups, the… So you made a statement there, I just wanna make sure that I get it clear. If I figure out the set of all ways that I can map circles, and spheres, and three-dimensional spheres, etcetera, into a space, I have not specified its topology completely, but I have specified its homotopy type topology completely, which is a slightly weaker thing. Is that right?
0:51:12.3 ER: That’s right, yeah.
0:51:13.5 SC: Okay. So do we know the answer to the general question of how to completely specify the topology of the space?
0:51:22.3 ER: Right. The classical approach to that… And it’s kind of a tautology, but let’s imagine it’s a theorem as opposed to a tautology. I mean, if you restrict to well-behaved spaces it’s a theorem, and if you take general spaces it’s kind of a tautology, but the classical approach to this is exactly what you’re saying. You describe what’s called the nth homotopy group as the collection of maps from the n-sphere into your space, and then because it’s a group, you need a composition operation, which for two spheres I can describe the composition. So if I have a map from a two-dimensional sphere into a space, and a map from another two-dimensional sphere into the space… And these maps are based so there’s the north pole, and they send them to the same point in the space.
0:52:12.9 ER: You could imagine taking another two-dimensional sphere and then collapsing the equator to a single point. So if you had a balloon and you collapse the points on the equator, you sort of squeeze at them carefully so it doesn’t pop and so now you have a single point. Now, what you look like is something that’s called a bouquet of spheres, it’s two different spheres that are glued together along the point where the equator used to be, and you could sort of map from that into the space because you have two different maps into the space that have a common point, and sort of the combination of that operation is how you define the composition here. And so there’s an analog of that in all dimensions, so this is the classical approach to saying, what is a space algebraically? What are some algebra stuff that tell you everything that you would want to know about the homotopy type of the space? But a modern approach goes back to the idea of the fundamental group, but replaces it by something called the fundamental infinity-groupoid, and since I promised I would tell the listeners what an infinity-groupoid is, let me do it now. So I like this because this feels like a much more natural way to describe the space and again, the fundamental groupoid, depending on your point of view, either it’s a theorem or it’s a tautology, really does capture the full homotopy type of the space.
0:53:33.1 ER: So what is it? So it’s some algebraic structure where I’m gonna start with the set of all points in the space. So I’ve forgotten the topology, I’ve forgotten about distances and stuff, I’m just remembering the set of points, ’cause in algebra, I have sort of sets and stuff, I don’t have geometry. So I’ve just remembered the set of the points in the space, then what I’m gonna throw in is the data of every possible path between any points in this space. This is gonna be a very big thing, by the way, so we’re covering out every possible path that an ant could take between points in the space, plus the composition.
0:54:09.2 SC: And by the way, I gotta say that mathematicians use the word data in a different sense than physicists use it.
0:54:15.0 ER: Oh yeah. All of our sets are infinite, it’s fine. No problems here.
0:54:19.2 SC: When you say the data of a path, you mean whatever information is required to specify that path among the space of all the paths.
0:54:26.7 ER: Right, so we have all the points in this space, we have all the paths in this space. Now when we were thinking about paths before, when we were talking about the torus, we were talking about, we only wanna consider paths up to being essentially the same sort of continuously deformable. We’re not doing that here, we’re literally remembering every single path, every path as a distinct path, but we are also gonna remember data of these continuous deformations, so there’s a notion of path between paths. If a path is a map from an interval end of the space, this is a continuous map from a square into the space of interval times, and interval is a square, and so map from a square into space can be understood as a path between paths where one edge goes as one path where the other edge goes as the other path, and then the other direction, the other dimension is giving the path between paths.
0:55:16.7 ER: So we’re gonna remember all of those as well, all of these paths between paths, so these are called homotopies, and then there’s no reason to stop at two dimensions, so we could take three intervals product it together, that’s a cube and remember all of those maps into the space, these are paths between paths, between paths, and then we could take paths between paths, between paths, between paths, paths between paths, between paths, between paths, between paths, and all the way up and that goes to infinity, that’s the infinity-groupoid. So that data… This feels like it’s maybe not an improvement, but somehow…
0:55:44.3 SC: Well, it’s something.
0:55:47.5 ER: Somehow that algebraic structure both describes the full homotopy type of a space, which is a useful way to think about it, but also invites you to imagine a generalization to a different world that’s further removed from the geometry, where you could imagine some of these paths are no longer invertible anymore, these are one-way paths.
0:56:12.6 SC: Oh okay.
0:56:13.7 ER: You might not be able to go backwards for whatever reason, and this is now the world of infinite-dimensional category theory, and that’s really where I work.
0:56:21.0 SC: So which brings up a couple… There’s a couple of things I just gotta clean up there, you were on a roll, so I just wanted to let you keep going, but one thing is just to remind folks when you talked about the path between two paths, right? The cube… Sorry, the square that was a map into it, that path between two paths might not exist. If the two paths are not homotopically equivalent that there won’t be any path between them, so there’s some structure in the space of what paths between paths exist.
0:56:50.7 ER: Right, by what’s present and what’s absent. That’s a beautiful way to say it.
0:56:53.9 SC: Good, and the other one was… This is a little bit off-topic, but when you did the explanation of the sphere and you squeezed it down at the equator to get the bouquet, not only is the language very beautiful, but the visualization is very compelling, and one thing that always gets asked, how do you visualize the infinity-groupoid? Is that something that is necessary for you to do? Do you approximate infinity by two, or is there some other trick?
0:57:21.1 ER: It’s hard. [laughter] I don’t know, you imagine a little piece of it at a time, and then… I don’t know. Yeah, it’s hard.
0:57:30.8 SC: Okay, that’s fair. Completely fair. I basically give the same answer, I say, You don’t. You do… The two-dimensional or three-dimensional examples you can get, but at some point, you have to trust the equations you’re pushing around. And I don’t think we’ve quite elaborated the difference between a group and a groupoid.
0:57:48.9 ER: Alright. The difference between a group is… The example of a fundamental group, the elements of the group are actually the loops themselves we’ve fixed as prior-ly given data, the home base point for the ant, and then the only further data we record are the loops in the space. So in a groupoid, you don’t fix a base point, you allow different base points, so the different points on the surface of… In the space. And so now you have two levels of data, you have the collection of different points, and then you also have the paths between the different points.
0:58:31.0 SC: Okay, is this… Outside of the world of topology and homotopy is… Are there groupoids? I mean, groups, physicists use groups all the time, right? SU3xU2xU1 is the symmetry group of the standard model of particle physics. We never use the word groupoid unless we’re secretly mathematicians, so just in the abstract sense. Is there a difference?
0:58:51.9 ER: Yeah, sure, so your groups are all automorphism groups of some object and it’s a fixed object. So you’re thinking about automorphisms of R3 or automorphisms of R4 or R3 with a chosen orientation or something like that. So your groups were all automorphism groups of a fixed object. In a groupoid you have different objects, so there’s not just one object anymore, there are different objects, and it’s exactly the mini-object analog of a group.
0:59:19.5 SC: Okay, I see. That’s not so bad, an automorphism is just a map from the space to itself. Is that right?
0:59:23.9 ER: Yeah.
0:59:24.1 SC: Okay, good.
0:59:25.2 ER: Yeah, yeah, sorry.
0:59:27.9 SC: We all know what automorphisms are amongst our friends.
0:59:30.7 ER: No, of course. I’m sorry about that.
0:59:31.9 SC: So then… Okay, good, sorry there was a lot of clarifying questions, but then let’s get back to the punchline here. The infinity groupoid, the sort of topological sense of all the different paths that we can map into the space and the paths between the paths, and paths between the paths of paths. So if we knew the infinity groupoid of a space, we would know what?
0:59:54.0 ER: Everything.
0:59:55.3 SC: Everything…
0:59:56.2 ER: Well, everything, if you only care about the space up to homotopy. I mean if you’re willing to say that n dimensional euclidean space is the same as a point, there’s no difference whatsoever, then yes, you know everything about this space. Now, if you care about geometry or dimension or things like that, then this is not the right point of view.
1:00:13.2 SC: But that’s okay. But for the torus, when you said, let’s calculate the fundamental group, and we noticed that it looked like… So for the sphere, the fundamental group is just trivial, so it’s one element. For the torus, it’s two copies of the integers, so it’s basically two integers you just give me. Is there… How do I even express what the infinity groupoid of a space is?
1:00:38.5 ER: Right, so let me move back to the sphere, ’cause of the two sphere, ’cause it’s a little easier to describe here. So if we’re thinking about the loops in the two sphere or the paths in the two sphere, there’s kind of nothing interesting to say. If you have any two points on… Sorry, the two sphere is the ordinary sphere, if you have any two points on the sphere, you can connect them by a path, and there’s a sense in which all paths are the same. You could continuously deform any path from x to y into any other path from x to y. But now if we think about these two-dimensional paths between paths, there are fundamentally different ones, and this is really surprising. So in one dimension, all paths are somehow the same.
1:01:16.7 SC: On the sphere.
1:01:17.2 ER: But in two dimensions, paths can be quite different. So I wanna think about paths between paths, and so I should fix the two paths first. So let’s start at the… Let’s start at the North Pole and the South Pole. So these will be paths from the North Pole to the South Pole, and one of the paths I wanna take is the International Date Line, so somewhere through the Pacific Ocean. And the other path I wanna take is the prime meridian which is… I don’t know, it’s through England or something like that.
1:01:45.2 SC: Through Greenwich, yeah.
1:01:47.3 ER: Okay, somewhere else. There’s the Greenwich one and the Pacific Ocean. So those are both paths from the North Pole to the South Pole. Now, a path between paths is a continuous map from a square onto the surface of the earth that sends one edge to the prime meridian and the other edge to the International Date Line. And one of them is the one that would cover Asia, that would go east from the prime meridian to the International Date Line. And then the other one is the one that would cover the New World, so go west. And those are fundamentally different in the sense that there is no three-dimensional path, no path between paths between paths, that continuously deforms the one to the other. If you could pass through the core of the earth, you could do that. But we have to stay on the surface. It’s not allowed.
1:02:36.3 SC: Right.
1:02:36.6 ER: So in this sense, the fundamental group or the fundamental groupoid, that was sort of a one-dimensional thing, does not describe everything that’s going on on the surface of the sphere. But once we allow these higher dimensional things, we do get everything.
1:02:49.9 SC: And is your… Is the kind of day job that you’re involved in more actually calculating the fundamental… The infinity groupoid of this or that, or is it more proving theorems about properties of infinity groupoids?
1:03:03.7 ER: Right. So that’s a great question, ’cause those are both very active areas, there are a lot of researchers working on both problems. I don’t do the calculations myself, they’re very hard.
[laughter]
1:03:14.2 ER: I do kind of the theory side, but some of my colleagues work on the calculations.
1:03:19.1 SC: I’m sure it makes everyone feel good that you don’t really do the hard part, you’re just doing the simple part, [chuckle] proving theorems about infinity groupoids.
1:03:23.0 ER: Exactly.
1:03:25.5 SC: And it also… This discussion is wonderful because it does… Beneath the surface, in the language that you use, in the way you talk about it, the role of the maps between the different spaces really shines through. Just think of all the spaces you can invent in all the different ways you can map them in some sense. And that sneaks us up into the topic of category theory, which is not really our focus here, but I don’t wanna leave the audience completely bereft of category theory while we’re here. So how do we get from topology to category theory?
1:04:00.5 ER: Sure. Again, there’s lots of different roots in, but maybe the one that’s most relevant to this conversation, and this is kinda back to the conversation on philosophy that we started off with… So the fundamental theorem and category theory or somehow that’s expressing the core philosophy of category theory. This thing called the Yoneda Lemma, says that if you have any sort of mathematical object, it could be a topological space or it could be a vector space, or it could be a ring, or it could be a field or whatever, any sort of mathematical object, you can understand everything that you wanna know about it by considering the other objects of that same type. So other spaces or other rings or other fields, and the maps between them.
1:04:46.6 SC: Oh my goodness.
1:04:48.3 ER: Right, so what this is saying in the case of spaces is that if you have unknown space, x, and you’re trying to understand that space, so we don’t know sort of what its dimensions are, what its points there, we don’t know anything about it, a theorem in category theory, the Yoneda Lemma, says that you can completely characterize your unknown space, by considering the other spaces. So all these spheres and tori and other surfaces and whatever in all dimensions, and then the continuous maps from that end of your space, x. But what’s cool about that result… So we were using this idea in topology already to understand spaces, but what’s cool is it’s completely independent of the mathematical context. So the same theorem is true for rings. You can understand a ring by thinking about other rings and the ring homomorphisms between them, you can understand a group by thinking about other groups and the group homomorphisms between them. This works in any mathematical context whatsoever.
1:05:44.2 SC: It’s again, a little bit related to some ideas in physics, that we should always be talking about relations between different things, rather than intrinsic essences of things themselves.
1:05:56.3 ER: Yeah, absolutely, absolutely.
1:05:58.3 SC: And why is this called category theory? Let’s just bite the bullet and tell people what a category is, maybe.
1:06:03.9 ER: Sure. A category is kind of like a very general template for a mathematical theory. So Barry Mazur has this metaphor, it’s like something with nouns and verbs. So a category is given by a collection of objects, these are the nouns, and what you should think of here, are all the rings, all the possible rings, so the integers, and the rationals, and the polynomials and matrices and blah, blah, blah. So all the rings. And then, you also have the sort of functions between them. So in the case of rings, these would be functions that respect the addition and multiplication laws; in the case of spaces, these would be continuous functions. And sort of that totality of information, the objects like the spaces and the functions, the continuous functions or maps between them and their composition and so on, that’s a category.
1:06:52.8 SC: And what’s great about this is, every word you say makes perfect sense, and at the end, I’m left not quite knowing what the implications of these ideas are. And that’s where… That’s the devil being in the details. You’ve ascended to this platonic realm of wonderful abstraction, where there’s just things and maps between them. So how do you… What’s the usefulness? What’s the cashing out of this? What is the free market value of a good category theory?
1:07:16.8 ER: Right. One nice thing about category theory is, you can [1:07:22.1] ____ say when two categories are the same in a essential sense. So there’s a notion of equivalence between categories, and I’ll give you my favorite example. So there’s a category whose objects are vector spaces, which are something that are kind of fundamental in modern quantum physics. So a vector space is like a collection of vectors with vector addition and scalar multiplication, and then the sort of transformations between vector spaces are called linear transformations. So this is kind of a bread and butter category objects or vector spaces are the maps or transformations, which are some sort of functions between these vector spaces. Now, there’s another category that’s kind of a lot simpler to define. So the objects in this category are just natural numbers, so I know exactly how many objects there are, each natural number is an object, there are no other objects, that’s it. And now I need to say sort of what a transformation is and…
1:08:17.9 SC: So, sorry, the natural numbers are 0, 1, 2, 3, the positive… Non-negative integers.
1:08:20.8 ER: 0, 1, 2, 3. Absolutely. Okay, so now I need to say what a transformation or what an arrow is, from one number, let’s say five, to another number, eight. And what it is, is it’s gonna be an 8 x 5 matrix. So to make something a category, I need to tell you about the objects and the arrows between them, and that’s what I’ve done. The natural numbers are the objects, the matrices are the arrows, but you also need a composition law, so I need a way to take a 5 x 8 matrix and an 8 X 7 matrix and produce a 5 x 7 matrix, but there’s a thing for that, it’s called matrix multiplication. It’s an operation, it satisfies the axioms of category. So those are two very different sounding categories. On the one hand, I have this very abstract, very large thing of all vector spaces and all linear transformations between them, and then on the other hand, I have this kind of tori category with natural numbers and matrices, and those categories are equivalent. So in other words, you can think of the natural number as a stand-in for vector space, the number five corresponds to five dimensional Euclidean space.
1:09:28.2 SC: I was gonna guess that. I promise, yes.
1:09:31.1 ER: Yeah, and you can think of matrix as a stand-in for a linear transformation. So if you have vector spaces and you choose a basis, then you can use those bases to get a matrix of numbers that encodes the full data of the linear transformation. And so, in the mathematics department, we often teach linear algebra in two different tracks. There’s a computational linear algebra. If you’re gonna be the next founder of Google, you need to learn how to use and do these matrix operations and you’ll take that sort of course. Or there’s a theoretical linear algebra that’s taken maybe by more math majors. And on the ground, the subjects feel very different because, one, you’re learning a lot about matrices and reduction and operations, and on the other, you’re learning this theory about linear independence and bases and so on and so forth. But it’s the same subject, because these are equivalent categories.
1:10:23.4 SC: Okay, that is actually a very good example of a little useful bit of insight that you get from thinking this way. So just… I know that this has sort of already been said by you, but let me try to say it again to drive home this notion of a category, to people who don’t use it as a bread and butter. ‘Cause when you say a vector space, let’s just imagine… Let’s optimistically imagine everyone knows what a vector space is, they have in their mind, x, y-axis and little vectors. So vector space is itself a collection of things, the vectors. There’s an infinite number of vectors in the vector space, but the category is of vector spaces, so the individual elements of the category are the whole vector space, a two-dimensional vector space, a three-dimensional vector space, etcetera, and you’re thinking about the maps between vector spaces and then extra maps between the set of all vector spaces in the set of all integers and things like that. So it gets pretty un-visualizable pretty quickly, but that’s why you get paid the big bucks.
1:11:20.2 ER: Well, the visual is, you’re sort of zooming out, you’re really taking a bird’s-eye view of mathematics. The objects that group theorists would study, are really just little atoms inside of the category of all groups, and what’s fun is, if you’re a category theorist or a higher dimensional category theorist, really the categories themselves become very small. So in my work, I zoom out one other level and I think about categories whose objects are categories. Inside those categories are things like the category of vector spaces, and then inside a vector space is a natural vector space, which has uncountably, infinitely many vectors in it, as you point out.
1:11:53.9 SC: And so, at what point do we get to the infinity categories? If there’s an infinity groupoid, there must be an infinity category, right?
1:12:00.1 ER: Yeah, sure. I mean every decade, mathematicians invent more complicated objects to study and the universes where those objects live are categories with more dimensions of morphisms between them, and those are these infinity categories.
1:12:20.0 SC: I mean, just knowing that can you foresee what was gonna be invented in the next decade? What is the obvious thing to draw more arrows between?
1:12:27.9 ER: Yeah. I mean, what I’m hoping happens in the future is that we change our foundation system of mathematics, so it’s kind of more suitable to these complicated up to homotopy structures that we’re thinking about today.
1:12:41.0 SC: Well, maybe this is a good place to end up because in some sense… It’s kind of fun… You and I are both in the small group of people who just think it’s fun thinking about this stuff, but it’s also maybe a shift of perspective on what math is in changing what we mean by equality and equivalence and things like that, and so can you imagine that math is gonna look very different down the road when this really seeps in, is it kind of like a shift from classical mechanics to quantum mechanics in some sense?
1:13:15.8 ER: Yeah, I think so. I think we maybe see glimpses of it today, but I think every living mathematician would be very surprised by 22nd century mathematics, and I hope to be around to see some of it.
1:13:32.1 SC: Well, it’s very interesting to read… There’s a wonderful interview with you in Quanta Magazine, and one of the interesting things you’re doing is writing books, maybe you count them as textbooks, but anyway, technical mathematical books where… Correct me if I’m saying this wrong, but you were just as interested in re-proving known theorems in better ways as improving new theorems, which is supposed to be the typical thing mathematicians are paid to do.
1:14:00.1 ER: There’s this… Bill Thurston, who is this wonderful topologist/geometer, drew attention to kind of the role that mathematicians need to play in making mathematics understandable to humans. Because something has been proven that means it’s true, which… A nice thing about mathematics is the theorems that were proven 2000 years ago are equally true today, but that doesn’t necessarily mean that it’s understandable by somebody who wants to build on those ideas and use them to do something else. So I think it is worthwhile to do a bit of tidying up and repackaging, streamlining… A wonderful success in the history of mathematics is these cutting edge discoveries that were very controversial or inconceivable to somebody 100 years ago are now stuff we teach undergraduates in their first and second year and…
1:14:53.4 SC: True. I mean, there was a lot of controversy over calculus, right? That was considered hard.
1:14:58.5 ER: Right. These pinnacles of human achievement are now something that thousands and hundreds of thousands of students are learning every single year, and I hope we continue to progress in that way.
1:15:08.0 SC: We should check back. 30 years from now, we’ll have you back on the podcast, and we’ll check to see whether or not Category Theory is taught to at least undergraduates, at least first year students in the math classes so that’s something to look forward to. Emily Riehl, thanks very much for being on the Mindscape podcast.
1:15:20.8 ER: Great, thanks for having me.
[music][/accordion-item][/accordion]
Pingback: Sean Carroll's Mindscape Podcast: Emily Riehl on Topology, Categories, and the Future of Mathematics | 3 Quarks Daily
Thanks for a great episode. As a mathematician working in a completely different area of mathematics I have always found homotopy fascinating. This episode reminded me of my student years.