∞-Category Theory for Undergraduates

At its current state of the art, ∞-category theory is challenging to explain even to specialists in closely related mathematical areas. Nevertheless, historical experience suggests that in, say, a century’s time, we will routinely teach this material to undergraduates. This talk describes one dream about how this might come about — under the assumption that 22nd century undergraduates have absorbed the background intuitions of homotopy type theory/univalent foundations.

[jontedeakin]

it's *my* sleepover and *I* get to pick the movie

[enderwiggins]

This is also really interesting to me from a pedagogical standpoint; tying it to paradigm shifts in physics in the twentieth century, It's abstractly kind of funny the _lack_ of emotional impact a lot of quantum mechanics has on undergraduates. When we studied tunneling through 1D stepwise potentials, no one was flipping the tables in disbelief; and when we studied the derivation of the uncertainty principle the vibe was very "okay neat, but I kinda already knew this intuitively". I think my generation of college students has been fed a cultural diet of quantum weirdness so much that now we're unfazed by it. Maybe this is the future of ∞-category theory, that proofs that now seem so wild will seem very intuitive for college students in a century.

[columbusmyhw]

"I might turn off the recording, if I can figure out a way to do that" is, in retrospect, a pretty funny way to end a video

[miguelamaral]

Well as a current undergrad, I definitely did not catch everything you talked about, but awesome stuff, I look forward to learning more in the future. Cheers!

[lukasjuhrich]

For anyone looking for this as well, the result she mentions regarding „Indiscernibility of identicals“ most likely is M.Hofmann and T.Streicher: “The groupoid model refutes uniqueness of identity proofs” from the 9th annual IEEE Symposium on Logic in Computer Science. DOI: 10.1109/LICS.1994.316071.

[CosmiaNebula]

Well, I look forward to having you as my professor in 200 years.

[Alberiana]

Thank you for this! Finally got to know what homotopy type theory might be about.

[tobyaldape]

Amazing introduction to HOTT! I loved the color coding of different "types" of objects. Helpful for math as well as programming. Keep up the good work!

[kaushaltimilsina]

Nice introduction to the ideas, especially to me since I am currently studying Programming with Categories and will be studying homotopy type theory. Homotopy type theory and Category theory blended together seems like a great way to implement many ideas from pure mathematics like Homotopy theories on Schemes (which I can sort of speculate and try implementing in haskell) and many other ideas. Such ideas will probably allow great representation power (condensed representation and computation) and I speculate that in the future computation will use many ideas from pure mathematics. Also, the coloring is the really cool; it's like a really fun book. Thanks for sharing a recording of the talk. I first came across infinity groupoids when watching discussions of Wolfram Physics Project on youtube, and they were using the idea for paths in spacetime in the discussions-viewing spacetime from Homotopy Type Theory- who observes what and who can prove certain equivalences (who = which observer). It was quite fun to watch.

Question at 16:56 is asking about if context morphism capital Gamma can be regarded as a formal grammar.

I believe yes, I think there could be an abstract formal language grammar Gamma of which words are tuples of types. It might exist only theoretically as such formal language is extremely expressive and also contains words which are not computable. An example of that could be a type being a Turing Machine that ends.

I think this question was extremely interesting, maybe just not formulated properly.

[jsmdnq]

The big issue in modern mathematics is that there are so many different explanations and languages for the same thing. Even advanced mathematics has this issue. EVEN mathematics that are suppose to solve the problem end up contributing to it. It's much like the issue, say, with programming languages. There are thousands of them and they all purport to solve some problem but end up just adding another language to the bunch and since it actually lacks something in some other areas it never replaces everything.

Category theory is essentially a unifying language that could be the theory of everything but people need to start trying to optimize and prune it and create a curriculum for all levels that are based in category theory. This way we all have the same language to use to describe things and we learn it from the start.

I mean, really there is only one language with many sub-languages. The problem is there is a ton of overlap and a ton of isomorphic and equivalent structure... so much so we need a whole damn new language to get a handle on it(e.g., category theory).

[kitconnick]

I didn't realise you had a channel! I remember your Numberphile videos on the marriage matching up problem, and it was always one of my favourites, you were always wonderful to listen to! Now I'm 4th year at Uni this is an interesting video in a different way. Thanks for this great and helpful video, look forward to looking at your other videos :)

[ymaysernameuay]

I don't know how I got here but I'll stick around.

[sebastianmullerbalcazar]

totally agree...actually thinking conceptualy about relations make absolute sense to start understanding....everything else.

[miguelamaral]

41:17 Wow okay, beautiful picture to visualize how two proofs of the same identity type may not be equal. What then could be an interpretation of the "holes" that are preventing "homotopy" (keeping in analogy) between proofs in type theory? It is interesting to think of what could be fundamentally different about two proofs about same thing.

[AliceB]

I just came back from work and apparently didn't close youtube or turn of my pc.
I just turned on my screen 50 minutes in and have no idea what you're talking about but it sounds very interesting!

[defamationlaw]

Love watching you teach Emily. Hello from Oz

[NNOTM]

Re: it being surprising that path induction works - I think it's not at all surprising in intuitionistic type theory, where our introduction rule only admits refl. It becomes surprising though once you have the homotopical interpretation and allow non-reflexivity paths.

[Apocalymon]

Hmm, I wonder if higher category theory can be used to advance the Wolfram Hypergraph Physics Project. Have you considered contacting Jonathan Gorard to discuss the application of category theory to the physics project?

[keithwynroe]

Fantastic talk, thanks so much! Do you think this work of putting math on different foundations will be mainly valuable from a “different perspective” perspective (e.g makes it easier to engage w/ topics like infinity-categories more readily), or do you think it’s also in some way “better” than the ZFC foundations?