Arcadian Functor

occasional meanderings in physics' brave new world

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Marni D. Sheppeard

Saturday, January 05, 2008

Riemann Rekindled II

It's wonderful to see the GRT lectures reach the topic of degroupoidification and homology. The dual groupoidification process would take a vector space to a groupoid, which might be a one object groupoid. An instance of such a process might be the exponentiation of a Lie algebra to its Lie group. Dually, multiple logarithms are associated to many object degroupoidifications, as we see with the MZV algebras.

Now a while back the Everything Seminar set off a series of posts on categorified sums, including goodies like

$-1 = 1 + 2 + 4 + 8 + \cdots$

so we expect Euler's relation may be written in many ways, such as

$\textrm{log} (-1) = i \pi = \textrm{log} (1 + 2 + 4 + 8 + \cdots)$

which I guess is a definition of $\textrm{log} (-1)$, or maybe of $\pi$, which turns up in the Riemann zeta function for integral arguments. I wish I could play this game all day, but alas, the restaurant is busy...


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