51,270 reputation
15193460
bio website math.berkeley.edu/~qchu
location Berkeley, CA
age 24
visits member for 5 years, 8 months
seen 15 hours ago

I am a third-year graduate student at UC Berkeley. I'm interested in the interplay between homotopy theory, quantum field theory, manifold topology, and higher category theory.


1d
comment $\pi_8(S^5)=\pi_8(SO(6))=\mathbb{Z}/24$?
@Yingfei: $\pi_8(S^5)$ is finite (e.g. by work of Serre) so the map to $\mathbb{Z}$ is necessarily zero.
2d
comment Normal subgroupoid?
When you say "groupoid" here do you mean "semigroup"?
Jun
30
comment Does projective imply flat?
@Alex: I think you can only conclude that Tor is zero if one of the arguments is flat; to conclude this from one of the arguments being projective you already need to know that projective implies flat.
Jun
30
comment free action on product of two spaces
This is not even true if $G, X, Y$ are all finite.
Jun
30
awarded  Enlightened
Jun
30
comment What is the reverse mathematical strength of the fundamental theorem of algebra?
Does Harvey Friedman's claim refer to the consistency of first-order PA or of second-order PA?
Jun
26
comment Existence of non-trivial characters on compact abelian group
Fair enough; maybe I should have said "this is an aspect of Pontryagin duality." The most important thing, I think, was to communicate that there was a common keyword the OP could use to find out more.
Jun
26
awarded  Good Question
Jun
26
comment Existence of non-trivial characters on compact abelian group
en.wikipedia.org/wiki/Pontryagin_duality#References
Jun
26
comment Existence of non-trivial characters on compact abelian group
Yes; in fact you can relax "compact" to "locally compact." This follows from Pontryagin duality.
Jun
23
comment Do representations of the universal enveloping algebra $\mathrm{U}\mathfrak{su}_2$ retain the Hopf algebra structure?
Oh, I misread your question. I think your truncation is not a Hopf algebra.
Jun
23
comment Do representations of the universal enveloping algebra $\mathrm{U}\mathfrak{su}_2$ retain the Hopf algebra structure?
If two algebras are isomorphic and one of them has an additional Hopf algebra structure then you can transport that structure along the isomorphism; this is called, as you might expect, "transport of structure."
Jun
18
revised Higher refinement of Seifert-van Kampen theorem on the language of hocolim
deleted 14 characters in body
Jun
18
answered Higher refinement of Seifert-van Kampen theorem on the language of hocolim
Jun
16
comment Categorical definition of infinite symmetric product
I don't see any reason to expect a monomorphism. The natural map $S_n \to \text{Aut}(X^{\otimes n})$ is often not a monomorphism, for example.
Jun
16
comment $\Omega X$-action on spectral $X$-bundles
@Jon: the data of an action of a grouplike $E_1$ space $G$ on an object of an $\infty$-category $C$ is precisely the data of a functor $BG \to C$. So the data of a functor $X \to C$ is precisely the data of a functor $B \Omega X \to C$, which is in turn precisely the data of an $\Omega X$-action on an object of $C$.
Jun
15
awarded  Nice Answer
Jun
14
comment $\Omega X$-action on spectral $X$-bundles
$X$ needs to be a based and connected space.
Jun
14
comment What are algebras for the little n-balls/n-cubes/n-something operads exactly?
$E_1$ algebras in a symmetric monoidal (higher) category themselves form a symmetric monoidal (higher) category, and one way to define an $E_2$ algebra is that it is an $E_1$ algebra in $E_1$ algebras. This generalizes: an $E_n$ algebra is an $E_k$ algebra in $E_{n-k}$ algebras. So by induction, you're just repeatedly passing to the $E_1$ algebras in what you had before.
Jun
14
comment What are algebras for the little n-balls/n-cubes/n-something operads exactly?
@Mark: it just means I can't describe them at a fixed category level, like I did for $n = 2$ where I could just use groupoids. For example, you might've expected that I can describe the $E_3$ operad as an operad in $2$-groupoids or something, but that's not true. And of course the compatibility can be made precise in general: it's made precise by the $E_n$ operads! This may seem not very explicit but that's the price to pay for doing the homotopically correct thing. I don't really know what you mean by "algorithm" here but here is one way to think about it: (cont)