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Qiaochu Yuan's user avatar
Qiaochu Yuan's user avatar
Qiaochu Yuan's user avatar
Qiaochu Yuan
  • Member for 15 years, 2 months
  • Last seen this week
  • Oakland, California, USA
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Category theory sans (much) motivation?
If you wanted to free yourself from thinking that morphisms are functions, a much easier choice is the category of sets and relations.
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Category theory sans (much) motivation?
A little off-topic, but if you want a lot of examples in algebraic geometry you should take a look at <a href="books.google.com/… table of contents.
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What is the size of the category of finite dimensional F_q vector spaces?
Here's a comment that might help somebody else: if A(x) = a<sub>1 x + a<sub>2 x^2 + ... is a generating function, then Q is the linear transformation that sends (a<sub>1, a<sub>2, ...) to (A(q), A(q^2), ...).
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Category theory sans (much) motivation?
For what it's worth, I never had an "algebra sucks" phase because for me, at least in high school, the big motivation for algebra came from number theory, and I've always been interested in number theory.
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Is every finite group a group of "symmetries"?
That argument's a standard trick: if you start with an inner product (u, v), define a new one by (u, v)' = 1/|G| sum (gu, gv). This inner product is G-invariant so the representation is orthogonal with respect to it.
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What is the size of the category of finite dimensional F_q vector spaces?
Some noodling around in Mathematica suggests that the answer is something like 1/q prod (1 - 1/q^n).
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Definition of infinite permutations
As well as a nice presentation, a good notion of cycle decomposition, and so forth. The set of all bijections from a countable set to itself, on the other hand, is terrible; one can imagine a bijection which, for example, encodes Chaitin's constant.
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What is the Hilbert class field of a cyclotomic field?
I think you need more reputation to do that.