That said, I like your idea of using the geometric series in categories with products and sums to get negative objects, with

-X = (X – 1)(1 + X + X^2 + …)

but it only seems to work if we can solve the equation Y + 1 = X for Y. Even then, it looks a little rocky: with FinSet and X = 1 = {0} you get

-1 = (1 – 1) (1 + 1 + 1 + …) = 0

which feels like a reflection of the pole at 1. It would be nice to patch this, but I think the tricks involved will be fundamentally algebraic, not metric or analytic. Just guessing, though: I haven’t had any luck defining negative sets yet…

]]>I noticed that in Sums of divergent series posting the first sum was nothing but a 2-adic representation of -1 as 1+2+2^2+2^3+…. More general representation is (p-1)(1+p+p^2+…). Apparently they had not noticed the connection!

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