Jul 1, 2026 · 59 min · 12 segments
Higher Category Theory, Homotopy & AI in Math 00:50 Introduction to Higher Category Theory 02:37 Understanding Homotopy Theory 09:44 Exploring Category Theory Basics 15:52 Limitations and…
Emily RiehlGuest
Thorsten AltenkirchHostDeniz SarikayaHostMaybe can you say a few words to, about homotopy theory and what it is and why it matters?
Uh, right, so, you know, there's this fundamental mathematical question, uh, maybe philosophical question of when is one thing the same as another thing? And there's different ways that you can answer it.
This is, uh, Leibniz's indiscernibility of identicals or identity of indiscernibles.
So, you know, two things are the same if and only if they have exactly the same properties.
Um, but that's not really how, uh, mathematicians use this notion of being the same.
Um, so to a, so to a category theorist, if you have two objects that belong to the same category and they somehow have the same shape, um, the, the technical term is isomorphic, the etymology is roughly that, sort of same shape.
Um, meaning, um, from the point of view of all the other objects in the category, they look the same, then they're the same even though they're not literally equal.
Um, so there are examples, you know, in the group theory you'll learn about different presentations of groups that end up having exactly the same structure, and even though the presentations are different, the groups are isomorphic, and a group theorist would consider them the same.
Um, you know, in linear algebra, any, you know, finite dimensional, any 10-dimensional vector space is considered the same as any other 10-dimensional vector space-
though depending on choice of basis, you know, you might need explicit isomorphisms to identify them.
Uh, s-so, you know, that's sort of one perspective of, um, what sameness means, but, uh, that's often not even flexible enough.
Um, so this, you, you, you think the, the Leibniz principle suggests, um, uh, a f- more intentional equality, so to say.
Uh, and, um, I'm, I'm, I'm actually, I'm big fan of this Leibniz principle be-because, um, how do you actually, how do you actually, uh, describe the difference o-of two things which you, which you call isomorphic? Uh, you, you, you have to observe the internal structure, and in my, in my world, you're not even permitted to do this, right? So, so this principle of, uh, equality of indiscernibles is actually valid in, in, in type theory, in, in, in, in, in, in Ho- in HoTT, right?
Maybe can you say a few words to, about homotopy theory and what it is and why it matters?
Uh, right, so, you know, there's this fundamental mathematical question, uh, maybe philosophical question of when is one thing the same as another thing? And there's different ways that you can answer it.
This is, uh, Leibniz's indiscernibility of identicals or identity of indiscernibles.
So, you know, two things are the same if and only if they have exactly the same properties.
Um, but that's not really how, uh, mathematicians use this notion of being the same.
Um, so to a, so to a category theorist, if you have two objects that belong to the same category and they somehow have the same shape, um, the, the technical term is isomorphic, the etymology is roughly that, sort of same shape.
Um, meaning, um, from the point of view of all the other objects in the category, they look the same, then they're the same even though they're not literally equal.
Um, so there are examples, you know, in the group theory you'll learn about different presentations of groups that end up having exactly the same structure, and even though the presentations are different, the groups are isomorphic, and a group theorist would consider them the same.
Um, you know, in linear algebra, any, you know, finite dimensional, any 10-dimensional vector space is considered the same as any other 10-dimensional vector space-
though depending on choice of basis, you know, you might need explicit isomorphisms to identify them.
Uh, s-so, you know, that's sort of one perspective of, um, what sameness means, but, uh, that's often not even flexible enough.
Um, so this, you, you, you think the, the Leibniz principle suggests, um, uh, a f- more intentional equality, so to say.
Uh, and, um, I'm, I'm, I'm actually, I'm big fan of this Leibniz principle be-because, um, how do you actually, how do you actually, uh, describe the difference o-of two things which you, which you call isomorphic? Uh, you, you, you have to observe the internal structure, and in my, in my world, you're not even permitted to do this, right? So, so this principle of, uh, equality of indiscernibles is actually valid in, in, in type theory, in, in, in, in, in, in Ho- in HoTT, right?
The rest of this transcript — segmented and speaker-labeled, so you land on the exact moment something was said
Search every transcript — by keyword, by phrase, or by meaning, across every show Radar indexes
Trends — what is surging across podcasts, measured against its own baseline
Alerts — when a name you follow appears in a newly indexed episode
No account is needed to search Radar.
