48,846 reputation
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bio website math.berkeley.edu/~qchu
location Berkeley, CA
age 24
visits member for 5 years, 6 months
seen 4 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.


15h
awarded  Nice Answer
Apr
17
awarded  Notable Question
Apr
13
comment Maps to the group completion
Surely being an H-space isn't a sufficient condition to admit a delooping?
Apr
13
comment k-linear abelian categories which are not categories of modules
On the other hand, that does suggest how to adapt the above construction to $B'$. We take the subalgebra of $B'$ consisting of sequences $a_i \in M_i(\mathbb{F}_2)$ such that $a_i$ is eventually a scalar and $\lim_{i \to \infty} a_i = a_1$. That might work. But it can be ruled out by removing $M_1(\mathbb{F}_2)$ from the product... and to rule out similar-looking constructions let's maybe only take the product over prime $i$.
Apr
13
comment k-linear abelian categories which are not categories of modules
@Eric: that does not work for $B$. The subalgebra $A$ you suggest is the subalgebra of sequences $a_i \in \mathbb{F}_2$ such that $\lim_{i \to \infty} a_i$ exists, and as such it has an extra map $A \to \mathbb{F}_2$ not factoring through the distinguished finite quotients of $B$, namely $\lim_{i \to \infty} a_i$ itself. The induced map $A \to B$ of Boolean rings corresponds topologically to the natural map from the Stone-Cech compactification of $\mathbb{N}$ to its one-point compactification.
Apr
13
revised k-linear abelian categories which are not categories of modules
deleted 12 characters in body
Apr
13
revised k-linear abelian categories which are not categories of modules
deleted 12 characters in body
Apr
13
revised k-linear abelian categories which are not categories of modules
deleted 12 characters in body
Apr
13
answered k-linear abelian categories which are not categories of modules
Apr
13
awarded  Announcer
Apr
13
comment k-linear abelian categories which are not categories of modules
Also, in the question, by "equivalent" do you just mean as $k$-linear categories or do you mean as $k$-linear categories together with a choice of fiber functor?
Apr
13
awarded  Enlightened
Apr
13
comment k-linear abelian categories which are not categories of modules
The result you cite (with "comodule" replaced by "finite-dimensional comodule") requires crucially the hypothesis that $C$ is essentially small, or else the endomorphism algebra of $U$ can fail to be a set. For an explicit counterexample, take the category of finite-dimensional $\text{Ord}$-graded vector spaces. $\text{End}(U)$ has an idempotent for each ordinal given by projecting to the subspace indexed by the corresponding ordinal.
Apr
9
comment How do I apply the Boolean Prime Ideal Theorem?
@Manny: that is a proof that the method ultimately won't work, but it's not an explanation of why I can't immediately adapt this method to prove the corresponding statement for maximal rather than prime ideals.
Apr
9
comment How do I apply the Boolean Prime Ideal Theorem?
So I guess the reason this method will not work to prove the corresponding statement for maximal rather than prime ideals is that maximality is not expressible in the first-order language of rings?
Apr
9
awarded  homotopy-theory
Apr
9
comment What's the cardinality of a higher category?
@Simon: yes, of course I want the obvious finiteness conditions. I would hesitate to call this number "Euler characteristic," though; to my mind the Euler characteristic is set up to have nice behavior with respect to cofiber sequences rather than fiber sequences.
Apr
9
comment Is the infinity-groupoid of a finite CW complex finitely-presented?
Another way to say it: a presentation of a group is a description of that group as the cokernel of a map between free groups. Keeping in mind the analogy to free resolutions of modules, we might say more generally that "presentation" means "description of an object as a colimit of free objects." And a CW decomposition is precisely a description of a space as an iterated homotopy pushout of spheres (which themselves are iterated homotopy pushouts of points).
Apr
9
comment Is the infinity-groupoid of a finite CW complex finitely-presented?
...new "generators" in the sense that they may (before we kill them again) give rise to higher homotopy and new "relations" in the sense that they themselves kill homotopy.
Apr
9
comment Is the infinity-groupoid of a finite CW complex finitely-presented?
This is a model-independent statement. Of course it depends on what the OP means by "free $\infty$-groupoid on a finite family of generators," but I think this is a reasonable interpretation. Think first about how one presents a group $G$ in terms of generators and relations. Topologically generators correspond to $1$-cells and relations correspond to $2$-cells. So in fact a presentation describes a CW complex of dimension $2$ (the presentation complex). This is the beginning of a presentation / CW description of $BG$, and one now needs to add $3$-cells, etc. These are simultaneously...