Sean Carroll's Mindscape: Science, Society, Philosophy, Culture, Arts, and Ideas - 146 | Emily Riehl on Topology, Categories, and the Future of Mathematics

Episode Date: May 10, 2021

"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. Johns Hopkins web page Google Scholar publications Quanta profile Wikipedia Twitter

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The podcast is old enough, I can now say over the years. Wow, 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, 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 want to talk about is math. We haven't talked about math as a pure subject that much here on Mindscape. We did talk to Stephen 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 realm. of very pure mathematics. So that's what we're doing here today. Pure mathematics, no equations, but we're going to try to talk about the concepts that you come across at the most elevated realms of mathematical thought.

Our guest today is Emily Real, 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 going to be talking about is topology.

Topology is the study of the properties of mathematical spaces that are invariant when you smush them, when you smoothly deform them a little bit, right? 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 want to 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. And we build these mathematical algebraic structures by asking the question, how do we topologically characterize these kinds of spaces? So we're led from topology into algebra. And Emily's going to take us there. We're going to discuss things like homotopy, groups, rings, groupoids, all these words. that you're not supposed to have necessarily been familiar with already. We're going to 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, you know, 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 would 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. You know, thinking like a mathematician gives you new handles on the world just like thinking like a physicist does. And so I think it's going to be fun. Let's go. Emily Real, welcome to the Mindscape Podcast. Thank you. You know, I know that as a physicist, as a theoretical physicist, you think about the universe and the many worlds of quantum mechanics and so forth, I'm often asked, like, what is the point of this? You know, are you making a better cell phone or are you carrying cancer? Like, why are you doing this? And I have my own answers. But as a mathematician who works on topology, category, things like that, you must also get this. So what is your, do you have a favorite answer to that kind of question? Sure. I mean, my dad loves to ask me what the practical applications are of the math I love to think about. And I think he knows that he's kind of needling me because, you know, that's just, it's just not the point for all mathematicians. I mean, for some, of course, it is. You know, 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

being and that's how I or that's what inspires me to be a mathematician yeah good no I think I'm not going to say that's the right answer because like you say different people have different answers but that's analogous to my answer and people ask me about looking for the Higgs boson and so many my physics friends will say like well it might someday 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 because we want to find out what's going on. And if there's an application someday, that'll be a benefit. 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, you know, there's realism versus non-realism, platonism versus, I don't know, anti-platatism.

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? I certainly don't know as much about it as some of your other guests, but I agree with that instinct. I mean, the things that I think about and I dream about and I talk about with my friends feel very real to me. And, you know, I don't expect that I'll kind of trip over one on my walk to campus. But, you know, they feel as real to me as, I mean, as anything else. I mean, I guess as I understand it, that, you know, a table that seems like a solid, real thing is not really real either. So, yes, I subscribe to Platonism.

Well, yeah, I mean, there's certainly some structure to math, right? Like, we all agree, you know, given the axioms, 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 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, you know, what a mathematical object really is is determined by the role that it plays within mathematics.

So, for example, you know, you could ask what are the natural numbers, you know, 0,1, 2, 3, 4, 5. 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, you know, natural numbers are something you can recursively define functions on. If you have, if you're trying to define a, you know, sort of a sequence of points or a sequence of real numbers. What you, a strategy, you know, Fibonacci sequence, for instance, is a famous sequence.

One, one, two, three, five, eight, 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 like that's what the natural numbers are. They are sort of the thing that you can recursively define functions on. Quite how that fits with plaintiffism is it entirely clear to me. Yeah, no, no, no, me neither. I did have a guest a while ago, James Leideman, 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

relating between them. Great. It sounds like he's a category theorist. It could be. Aren't we all a category theorist or 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? I mean, 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? Great. So topology is the study of spaces, both, you know, kind of physical, geometrical spaces that we move in in the

world, but also spaces that you might imagine that, you know, have some resemblance to, you know, Euclidean space, three dimensions, two dimensions, one dimension, but are somehow more exactly. exotic. So a topological space could be, you know, 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, 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 it and all 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. The topology is a realm to do that.

Can I just ask very quickly, when mathematicians use the word space, obviously doesn't simply mean the three-dimensional space in which we live. What's the difference between space and set? 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. 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 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. Okay.

And so a line is a space, a circle is a space, a sphere. So for mathematicians, the sphere is just the surface. of the sphere, right? It's not the interior, like you said, the two-dimensional sphere is the surface of the ball. And then you could have like a hundred-dimensional sphere or whatever. So there's a lot of spaces we have to play with. You can have Mobius spans. So that's a space that's kind of like, I mean, it's kind of like a cylinder. So it's like a toilet paper tube. But imagine you cut the toilet paper tube, you know, 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 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 or the holder. Do you have one of those Klein bottle sculptures? I have a Klein bottle sculpture.

I don't, but, right? So 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. It's kind of like the surface of a sphere. Let me describe how you would build a Klein bottle. So 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, kind of a simpler construction that isn't the Klein bottle as you could just sort of 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. The Torres. We can use the word, it's okay. Okay. We can use the word, torus.

So if I want to get a Klein bottle, it's the same idea. I'm going to 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 sort of zipping up the going counterclockwise around one of the circles and clockwise around the other circle, but I want to reverse the orientation on one of the circles before doing this. Now, you can't do this in three-dimensional space. 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 and

there are these toy glass versions. And what's fun about it is there's 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 sort of fill it with jelly and the jelly is inside the torus. It's not outside. You know, until you bite it, you're not going to get jelly everywhere. 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 sort of flip it upside down and flip it right-side up again, it's going to get all over your desk because there's no inside and outside. And this is the stock and 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 and what it cares about. Exactly. Yeah, very loosey-goosey. And so, 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. Yeah, totally. So, 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 point 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. So, you know, imagine you have a factory and there's a sort of one dimensional track on the floor in the factory. So there's a, there's a strip essentially. And, you know, maybe it's a kind of a wheel or like a train track. You can move forward and backwards along that track. And we're going to have two robots positioned on the track. There's a red robot and a blue robot. So, well, let's actually start with one robot. So if you wanted to, if you had

one robot on a track, then you can describe its positions, you know, using just kind of a one, one dimension. It's like an interval. You can have the robot all the way at 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 because the robots are still on this one-dimensional track.

But actually, you know, you should use kind of 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. You know, one, you know, if these are physical robots, 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 the square with a 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. That's right.

But, you know, 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 of course, what you're trying to do with these robots on this track, is you could make a circular track instead of a, you know, straight track. And if you, a circle is also a one-dimensional object. You just sort of move around it going clockwise. There's kind of one direction of motion. But if you had two robots on a circular track, then the way you describe their configuration

space is you would take, you know, each robot is some arbitrary point in the circle. We have two circles, though. So we 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. Right. So the 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 going to make a diagonal cut that it's a cut that goes around sort of both loops in the torus once.

So you actually did this at home last night to test it in the way my toilet paper tube was made. You're an experimentalist. Good. How to seem exactly there. So it was easy to see that cut. It should be sort of a diagonal cut that moves once around in both directions. And if you do that, you're left with a connected surface. And that corresponds to the idea that if you have two robots on a circular touch,

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. Some of the most difficult and important decisions we make involve choosing people, whether it's choosing a PhD thesis advisor or choosing a prospective spouse or choosing who to hire for your company. I can't help you with the first two, but if you're looking to hire people, I can recommend Indeed. Indeed is the job site that makes hiring easy. You can post your ad, you can screen candidates, and interview them all on Indeed. Indeed gives you tools like Indeed instant match, giving you quality candidates whose resumes fits your job description immediately,

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at Indeed.com slash mindscape offer valid through June 30. Terms and conditions apply. Right, because it's good, because what topologists care about are the features that don't change by like small deformations of the space. So the fact that a space is

in two discontinuous, components like the original example versus just being one connected component, like the second example, that's the kind of thing that gets topologist very excited. 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. The famous thing, I bet many people have heard that to a topologist, a coffee cup is the same as a donut, right? Because they're both tourists. Right. Right. I mean, 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, you know, 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 tourist now, built out of clay and you have a solid coffee cup. You can define one of these homeomorphisms, this sort of one-to-one correspondence between the points and the Taurus 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. So, I mean, it's true that topologists consider Taurus and a coffee mug to be the same, but for like really, really, kind of obvious reason. They're just so 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 homotopia equivalence. And here are the ideas. Two spaces are homotopia equivalent if you can continuously deform one to another. And so let me give you some examples. So all of Euclidean space, you know, three-dimensional Euclidean space is homotopia equivalent to a single point. So you could imagine, I think of this Hombatopia equivalence is like the reverse Big Bang.

So you imagine three-dimensional Euclidean space, you know, 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. I realize the universe is probably not three-dimensional, but just bear with me here. Also, it doesn't have a center, but that's okay. We're going to, your method. And Orin-R-Nor does have it 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 going to do is I'm going to continuously move all the points in the universe back in a straight line to the center of the center of the universe. the universe to the origin. And they'll move kind of faster, slower, depending on how far 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 homotopia equivalence that feels more fun to me than the 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. They can be continued, 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. 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.

You know, all n-dimensional Euclidean space is also homotopia 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. You can imagine a homotopia equivalence that sort of, you know, shrinks sort of vertically, I guess, you know, and just leaves the thong from the pair of pants. And this is the, 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. Yeah, totally, totally. And this is very pervasive in mathematics. You know, mathematicians love trying to understand the sense in which things are the same,

and it's often a pretty loose sense. 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, right, which are very important from the early universe. None of the things I calculated turn 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, you know, so far it's been fun, homotopy, you know, smoothly deforming things into each other, but now we want to be a little bit more rigorous about it. We want to characterize,

we want to use this notion of smoothly deforming one thing into another to characterize the topology of different spaces, right? I mean, is in some sense our job, to sort of characterize all topological spaces? Yeah, exactly.

A way to, a classical problem, which is a good way to visualize this, is you can ask how many different surfaces are there, you know, sort of in this sort of very flexible sense, so not necessarily just. So for instance, we've described a few different surfaces. There's the sort of surface of a sphere. There's a surface of a torus. You can have many hold. The torus was the donut shape.

You can imagine having many hold donuts or like, you know, 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, you know, one donut with three holes in it. You know, that's another surface. And there's a classification question, you know, how can we tell that these spaces are really different? I mean, they seem different perhaps. Maybe I'm giving away my intuition, but they seem different. You know, how, but how do we prove?

that they're different because we've seen that these wildly different spaces are in some sense the same. Yeah. So how do we do that? So yeah. So the idea is, I mean, 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. Yeah. So 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 if, in a way that would give the same answer if the spaces were homotopia 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 invariants, then you know that they were different. Right. So let me go ahead. Well, I can give an example of this. Yeah, exactly. Do that. So what I'm going to 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, you know, imagine you're an aunt 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. Not the interior in either case, just the surface.

Not the interior in either case, right? The ant can't eat the donut or eat the sphere. The ant is sort of stuck on the surface. And I can't, like, jump off either. It's, you know, okay. So, right. So the invariant that I'm going to use to prove that these are different is something called the fundamental group.

That is, it's a, and what it's counting essentially. So in each, for convenience, I'm going to 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 Taurus. 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. So the subtlety there, as I said, the number of essentially different ways.

So, I mean, 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. Okay. So an example of two ways that are essentially different is we can imagine starting at our home-based 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. The short way around, basically. The short way around. Great. So that's one way the aunt could walk. Or the aunt could walk, let's say, are 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, you know, where the dough bit of the donut is, I could do that. I could sort of cut, cut through the bit that's not on the surface, but that's not allowed. So walking in the surface, once the ant decides it's going clockwise rather than counterclockwise,

those are different choices that the ant can make. In fact, they're homotopically inequivalent, right? Homotically incovalent, right? Continuously deform one into another. Right. And what's cool is you can actually enumerate all of the different choices. So another different trajectory of the ant could take is it could walk the, a 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, you know, 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. Right, right. So on a, on us, 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 a 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, sort of 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. so forth. And eventually all of the trajectories are deformable to just the ants 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. Which is wonderful because, you know, it took us on a little journey. I actually want to dig into this because on the one hand, we're not surprised that the sphere and the tourists are topologically different, but it was a bit of an effort there to actually show it, right? Even in that case, where we had things under control. And in doing that, you know, you stumble across, so you ask the question, how many different ways are there to sort of do this path

that returns to itself, a loop, right? A closed circle in this space. And in the case of the Taurus, you just uncovered this pair of integers, 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. Like, roughly speaking, mathematicians are either geometers and they like space or they're algebraists and they like equations, right? But an algebraic structure appeared here somehow. Right. I think, you know, this is just the tale of complexity in the modern world. Like, everybody has to be everything these days. So even if you really, your heart isn't with geometry, you have to use some algebra. So. Well, I want to, I want to just take advantage of your being here to dig into this a little bit more. I mean, we asked this question about how many ways you can draw a circle, roughly speaking, in the tourists. 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, right? And that's the beginning of algebra. Am I too dramatic there? Yeah, that's exactly right. So a more I mean, I've described this, in describing the fundamental group, I said, you know, let's imagine we have a home base point where they, and we're going to 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, you know, I don't, I don't know where it would be on the surface of a torus, for instance. So another way to think about this sort of algebraic structures, 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, you know, maybe the ant is tired and it takes a 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. So this is a composition operation on the path.

and the surface of a torus on any space, paths in any space. And really the sort of invariant that more 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 sort of infinite dimensional analog of a group, an infinity groupoid, which is a type of mathematical object we teach undergraduates, but an sort of infinite dimensional analog of a groupoid, which is a, you know, sort of frontier of research level mathematics. Right. When people turn to telehealth for weight loss, they're looking for real support. That's why more people are choosing orderly meds.com. Orderly meds connects you with real doctors and access to proven GLU-1 medications like semaglutide and terseptatide.

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Hey, everyone, it's Cal Penn. I'm the host of Earsay, the Audible and I Heart Audio Book Club. This week on the podcast, I am sitting down with Ray Porter, the narrator of Andy Weir's audiobook Project Hail Mary, massive sci-fi adventure about survival and science, and what happens when you wake up alone very far from Earth? I really had to make a decision because I caught myself getting that frog in my my throat and starting to get teary as I'm narrating some of these sections and it's like, okay, yo, yeah, yo, is this indulgent? And I really thought about it. I was like, no, at this point,

it would kind of be betraying the trust the author and the listener have in telling this story if I don't go through it. But there's places in this book that deeply emotionally affected me and I left it on the mic. That's great. Because it served the story. People will say like, oh my God, I cried at the end. It's like, yeah, dude, me too. Listen to your say, the audible. an IHeart Audio Book Club on the IHeart Radio app or wherever you get your podcasts. I think that if, look, if people are going to listen to this entire podcast episode, I want them to come away knowing what an infinity group boy is. I think that's going to 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 know, you mentioned the fundamental group, which is the collection of the circles we started with. So what's a group? Right. So a group is, I mean, a way to, I mean, this, like it, there's, there's this, there's this metaphor of the blind men and the elephant and, you know, 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, that's how I'm thinking about groups. So it's like really hard to know, know how to start. But. you know, 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 a composition operation. 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, you know, sort of very natural axioms. So if you're walking along a loop, you can. 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 sort of where you started. And a few sort of simple axioms like that. So it's kind of a, it's a stripped down version of, you know, the integers or something like that, right? Where the integers, you can add them together. So the integers are an example of a group,

right? Right. Intigers with addition is an example of a group. Absolutely. That's right. 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. Okay. And the physicists love group theory because symmetries are a group, right? Like rotations and translations and things like that.

Yeah, absolutely. So another perspective on groups, if we sort of move around the elephant, is a group is an aximatization of the symmetries of an object. So, 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 kind of flip your mattress occasionally, because I guess it's good for the life of the mattress. And so you might wonder, like, how many different ways are there to flip the mattress?

How many different configurations could the mattress be in? And, well, it's the mattress, one option is whatever the mattress is when you start. it. Then you could sort of rotate it head to toe. So you're sort of switching the head and the toe, but the top stays the same. That's one move. You could also flip it sort of 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. Yep. Or you could combine those two operations. And the effect of this is the head is now where the toe was and the toes were the head and the top and bottom surface has also. flipped. 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 composed to each other. So each of those flips is an 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 for 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 sort of 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. 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. Yeah, add one.

negative one, you get all the way through everything. So yes, the integers are cyclic group, which have a single generator, absolutely, but Klein 4 group is not. Right. So the integers may be to people who are not group theory, fissionados, are a nice little paradigm, but it's important that groups can be very different. So this is a finite group, this mattress flipping group, right? How many elements are there in the mattress flipping group? Four. Yeah. Four elements. Okay. Yeah. All right. And there's, so, I mean, these groups are super cool. And, you know, 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 the sides have the same length, all the angles are the same and so on and so forth. And, you know, there are infinitely many of these because there's a Heptagon and an octagon and an anonagon, you know, for any natural number N, you can get a plain figure with n sides. So you might wonder how many of these regular platonic solids are, you know, how many different shapes you can get by gluing together, say, triangles or gluing together squares or gluing together pentagons or, you know, 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 sort of 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. But clearly that's not what Plato did. I don't know, maybe somewhere like 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 kind of 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, so it's, sorry, its spaces are squares. there are six of them, and they are glued together along 12 different edges, and there are eight corners. And what I'm going to do is I'm going to 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 kind of 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,

be 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 tube, the same shape as the symmetry group of the octahedron and so on. Anyone who played Dungeons and Dragons knows about the Platonic solids because they have the dice, but they don't know about the duality relations. So now that's good. That's something else that they have in their bag now.

I'm only going to do this at great risk, but I figure, like, all right, while we've talking about groups, why don't we also explain rings and fields to the audience, since these are the other sort of algebraic structures that mathematicians love to throw around? Yeah, absolutely. So, I mean, what's funny is, you know, after a while, you kind of forget that these terms refer to other things. So, I mean, you know, when I say field, I mean, I definitely think about the mathematical field before. I remember that's like also the thing that's out the window. But, right, right.

So a group is describing a setting where you have, you know, collections. 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, you know, 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 sort of familiar for the integers. So you can understand there's a distributivity property that says if I add two integers together and then I multiply by something, it's the same as multiplying first and then adding.

And there are a few axioms like that. So what's cool about, I mean, you might ask like, why do we bother? Like 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, it's kind of interesting fact is there's a very deep analogy between the integers and polynomials. So a polynomial is like this is sort of the 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 sort of 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, 17 X squared plus pi X cubed because we can have real numbers as coefficients minus three. 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 sort of 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 in the real numbers. And the ring of integers are very quite similar as rings. They have a division algorithm. You can do sort of long division with polynomials like,

with rings. All the ideal elements are principles. And so, I mean, that's the sort of thing that's of interest to mathematicians are these kind of deep analogies between structures that are superficially quite different. Well, and what's the difference between a ring and a field? Because they're kind of similar. 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. So if I, in that, every addition, every element has an additive inverse. So if I pick my favorite integer, you know, 17, there's another integer, negative 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. Right. 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. You're sort of assuming you're not trying to divide by zero, that number 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. And does the existence of the number zero get you in trouble because it doesn't have inverse? Yeah, so in a field you have to treat zero as a kind of special case. So in a field, you have to, your zero and your one can't be the same. Otherwise,

the whole thing kind of collapses. And zero will not have a multiplicative inverse, but every other element has to. Got it. Okay. And so when we started, we started this little journey, this whole sidetrack, because we were thinking about the Taurus 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 invariants? 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. But, you know, a single group will not capture the full data of a topological space. I mean, really what you have to do is have introduced lots of different groups that are measuring lots of

different things. So the fundamental group tells you about sort of loops in the space and whether you can, if you sort of walk along a circular path in the space, can you fill that in with a disc? You know, could you, I guess I don't know how to say it any better than that. That's a good way to say it. Can you contract it? You know, yeah, if you have sort of a wire ring, is there, could you put sort of a soap bubble on it somehow so that the surface of the soap bubble lives within the space that you were talking about. So that's not possible if you're walking the short way around on a torus because the soap bubble would have to kind of cut through the dough,

and that's not on the surface. But it is on the surface of a sphere. You could just kind of paint over the disc. So that's measuring this kind of one-dimensional holes, I guess, is the area that's bounded by a one-dimensional sphere. But as we mentioned, there are kind of spheres in higher dimensions. So you could ask if you have a balloon inside your surface, can that be filled in with sand, sort of 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 sort of infinitely many groups, it does. 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. 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. Yeah, absolutely. 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 hyper rings or something like that? Sure. I mean, you know, algebra is a very flexing. subject. And, you know, to define what an algebra is in full generality, you need, you know, kind of the collection of elements that you're considering. And then you can specify arbitrarily many operations with arbitrary erities. And you can specify arbitrarily many rules between them. And, you know, the subject of universal algebra invites you to consider examples like that.

Let's invite listeners in the audience who are inclined in that direction to follow study universal alabas. But we're going to go back to the topology. and the groups, the, so you made a statement there, I just want to 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, et cetera, 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? That's right, yeah. Okay. So do we know the answer to the general question of how to completely specify the topology of a space?

Right. So a classical, the classical approach to that, and it's kind of a totology, but let's imagine it's a theorem as opposed to a topology. I mean, if you restrict to sort of well-behaved spaces, it's a theorem, and if you take general spaces, it's kind of a totology. But so the classical approach to this is exactly what you're saying. So you describe what's called the sort of enth homotopy group. as the collection of maps from the end 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 kind of north pole and they send them to the same point in the space.

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 sort of two different spheres that are glued together along the point where the equator used to be. And you could sort of map from that end of the space because you have two different maps into the space that have a commonplace. common point and sort of the combination of that operation is how you define the composition here.

Okay, good. 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 your listeners what an infinity group void is, let me do it now. So I like this because this feels like a kind of much more

natural way to describe the space. And again, the fundamental group void, 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. So what is it? It's some algebraic structure where I'm going to 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. Okay.

You know, because in algebra, I have sort of sets and stuff. I don't have geometry. So I just remember the set of the points in the space. Then what I'm going to throw in is the data of every possible path between any points in the space. This is going to be a very big thing, by the way. So we recover now every possible path

that an ant could take between points in the space. And by the way, I mean, I got to say that mathematicians use the word data in a different sense than physicists to use it. Oh, yeah. I mean, all of our sets are infinite. It's fine. No problems here. When you say like the data of a path, you mean whatever information is required to specify that path among the space of all the paths. Right. So we have all the points in the space. We have all the paths in the space. Now, when we were thinking about paths before, when we were talking about the Taurus, we were talking about, you know, well, we only want to consider paths up to being essentially essentially the same sort of continuously deformable. We're not doing that here. We're literally remembering every single path. Every path is a distinct path. But we are also going to remember

data of these continuous deformations. So there's a notion of path between paths. If a path is a map from an interval into the space, this is a continuous map from a square into the space. An interval times an interval is a square. And so a map from a square into space can be understood is a path between paths, sort of where one edge goes is one path, where the other edge goes is the other path, and then the sort of other direction, the other dimension is giving the kind of path between paths. So we're going to 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 producted 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, pass between paths, pass between, past, between past, between past, and all the way up, and that goes to infinity. That's the infinity group void.

So that data, I mean, this feels like it's maybe not an improvement, but somehow. Yeah, it's something. Somehow that algebraic structure, you know, 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 can imagine some of these paths are no longer invertible anymore. These are sort of one-way paths. Oh, okay. 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. So free is great, but only if it's useful. Free credit scores from some apps can

differ by as much as 100 points from your actual FICO score that 90% of top lenders use when you apply for a credit card, personal loan, car loan or mortgage. That can mean a higher interest rate, a bigger monthly payment, or worse. Denied. My FICO gives you your actual FICO score. The score lenders use straight from the company that created it. For the moments that matter, get the score that matters, your FICO score. Visit myfico.com and get started for free today. Well, so, which brings up a couple, there's a couple things I just got to clean up there. You're on a roll, so I just wanted to let you keep going. Sure. One thing is just to remind folks, would you just, talked about the path between two paths, right, the cube, or sorry, the square that was a map into it,

that 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 pads between paths exist. Right, by what's present and what's absent. That's a beautiful way to say it. Great. 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? It's hard. I don't know. I don't know. I mean, you sort of imagine a little piece of it at a time and then, I don't know. Yeah, it's hard. Okay. That's fair.

Completely fair. I basically give the same answer. I say you don't. You know, 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 group void. Right.

So the difference between a group is, so the example of a fundamental group, the elements of the group are actually the loops themselves. We've kind of fixed as priorly 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 group void, you don't fix a base point. You allow different base points,

so the different points on the surface of, or in the space. And so now you have kind of two levels of data. You have the collection of different points, and then you also have the paths between the different points. Okay. Is this, I mean, outside of the world of topology and homotopy, are there group voids? I mean, groups, physicists use groups all the time, right? SU3 cross SU2 cross U1 is the symmetry group of the standard model of particle physics.

We never use the word group void unless like we're secretly mathematicians. So just in the abstract sense, is there a difference? Yeah, sure. So your groups are all automorphism groups of some object, and it's a fixed object. So, you know, you're thinking about automorphisms of R3 or automorphisms of R4 or sort of R3 with a chosen orientation or something like that. So your groups were all automorphism groups of a fixed object. In a group where you have different objects.

So there's not just one object anymore. There are different objects. And it's exactly the many object analog of a group. Okay, I see. That's not so bad. And automorphism is just a map from a space to itself. Is that right? Yeah.

Okay, good. Yeah. Yeah, sorry. So. We all know what automorphisms are among the star friends. No, of course. So then, okay, good. Sorry, there's 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 group void of a space, we would know. what? Everything. Everything. I mean, 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 the space. Now, I mean, if you care about geometry or dimension or things like that, then, you know, this is not the right point of view. But that's okay. But for the tourists, when you said, you know, let's calculate the fundamental group, and we noticed that it looked like, so for the sphere, the fundamental group

was just trivial, so it's one element. For the Taurus, it's two copies of the integers. So it's basically two integers you just give me. How do I even express what the infinity groupoid of a space is? Right. So let me move back to the sphere because the two sphere, because 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 the, sorry, the two spheres 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. But in two dimensions, paths can be quite different.

So I want to think about paths between paths. And so I should fix the two paths first. So 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 want to take is the International Date Line, so somewhere through the Pacific Ocean. And the other path I want to take is the Prime Meridian, which is, I don't know, it's through England or something like that. Through Greenwich, yeah. Okay, somewhere else.

So right. So 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 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. So in this sense, the fundamental group, or the fundamental group, or the sort of one-dimensional thing does not describe everything that's going on on the surface of the sphere. But once we allow these higher three-dimensional things, we do get everything. And 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 group-oids?

Right. So that's a great question because 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. You know, I do kind of the theory side, but some of my colleagues work on the calculations. I'm sure it makes everyone feel good that you don't really do the hard part. You're just doing the simple part.

Yeah. Yeah. Yeah. Yeah. And it also, this discussion is wonderful because it does, you know, 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. You know, you know, just think of all the spaces you can invent.

and 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 want to leave the audience completely be roughed of category theory while we're here. So how do we get from topology to category theory? Sure. I mean, again, there's lots of different roots in,

but maybe the one that's most relevant to this conversation. And this is kind of back to this, back to the conversation and philosophy that, we started off with is, so the fundamental theorem in category theory, or somehow that's expressing the core philosophy of category theory, this thing called the Unatea 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 want to 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. Oh, my goodness. So, 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 are, we don't know anything about it. A theorem in category theory, the Yaneda Lama, 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 in D or space X. Well, okay. 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. 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. Yeah, absolutely. And why is this called category theory? Like let's just bite the bullet and tell people what a category is maybe. Sure. So I mean, 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 like all the rings, all the possible rings. So the integers and the rationales and the polynomials and the 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. 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. That's the devil being in the details, right? You've ascended to this platonic realm of wonderful abstraction,

where there's just things and maps between them. So what's the usefulness? What's the caching out of this? What is the free market value of a good category theory? Right. So, I mean, one nice thing about category theory is you can just 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 sort of 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, the maps are 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 from N. Okay. The natural numbers are 0123, the positive, non-negative integers.

Yeah. Absolutely. Okay. So now I need to say what a sort of transformation or what an arrow is from one number, let's say five to another number eight. and what it is is it's going to be an 8 by 5 matrix. Okay. 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 by 8 matrix and an 8 by 7 matrix and produce a 5 by 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, you know, very large thing of all vector spaces and all linear transformations between them. And then on the other hand, I have this kind of toy 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. I was going to guess that, I promise, yes. Yeah. And you can think of a matrix as a stand-in for 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 a mathematics department, we often teach linear algebra in kind of two different tracks. There's a sort of computational linear algebra. If you're going to be the next founder of Google, you need to learn how to do these matrix operations, and you'll take that sort of course. or there's a theoretical linear algebra that's, you know, taken maybe by more math majors. And this, on the ground, the subjects feel very different because one, you're learning a lot about matrices and reduction in operations, and then 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. Okay, that is actually a very good example of like a little useful bit of insight that you get from thinking this way. So just, I know that this is 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 in bread and butter. Because when you say a vector space, let's just imagine, let's optimistically imagine everyone knows what a vector space is, right? They have in their mind X, Y, axes and little vectors. So a vector space is itself a collection of things, right? 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, et cetera. And you're thinking about the maps between vector spaces and then extra maps between the set of all vector spaces and the set of all integers and things like that. So it gets pretty unvisualizable pretty quickly, but that's why you get paid the big bucks. Well, the visual is you're sort of zooming out. You're really taking a bird's eye view of mathematics. You know, the objects that, you know, group theorists would study are really just little atoms inside 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 inside of vector spaces, it's an actual vector space, which has uncountably infinitely infinitely many vectors in it, as you point out. And so at what point do we get to the infinity categories? There's an infinity group or there must be an infinity category, right? Yeah, sure. I mean, yeah, you know, as 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.

I mean, just knowing that, can you foresee what could be invented in the next decade? Like, what is the obvious thing to, like, draw more arrows between? 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. Well, maybe this is a good place to end up because in some sense, like this all, it's kind of fun. Like you and I are both in the small group of people who just think it's fun thinking about this stuff, right? But it's also maybe a shift of perspective on what math is, right, in changing what we mean by equality and equivalence and things like that.

And so can you imagine that math is going to 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? Yeah, I think so. I think, you know, we maybe see glimpses of it today, but, you know, I think, you know, I think you know, every living mathematician would be very surprised by, you know, 22nd century mathematics. And I hope to be around to see some of it. Well, I was very interested to read.

There's a wonderful interview with you in Quanta magazine. And one of the interesting things you're doing is writing books. I mean, 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 reproving known theorems in better ways as in proving new theorems, which is supposed to be the typical thing mathematicians are paid to do. Right. And, you know, there's this Bill Thurston, who is this wonderful topologist geometer, drew attention

to, you know, kind of the role that mathematicians need to play in making mathematics understandable to humans, you know. So, you know, because something has been proven, that means it's true, which, you know, a nice thing about mathematics is the theorems that were proven 2,000 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, you know, kind of tidying up and repackaging, streamlining. You know, a wonderful success in the history of mathematics is, you know, these cutting-edge discoveries that were, you know, kind of very controversial or

inconceivable to somebody 100 years ago are now stuff we teach undergraduates in their first and second year. True, right. I mean, there was a controversy over calculus, right? That was a, that was considered hard. Right. These pinnacles of human achievement are now something that, you know, thousands and hundreds of thousands of students are learning every single year. And I hope we continue to progress in that way. We should check back 30 years from now. We'll have you back on the podcast and we'll check to see whether 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. Real. Thanks very much for being on the Mindscape Podcast.

Great. Thanks for having me.