BCN Spring 2016 Workshop: Number Theory & K-theory
From zeta functions to Waldhausen's S-construction
Ponents
Joachim Kock
Resum
The classical Möbius inversion principle for arithmetic functions states that the zeta function is convolution invertible (with inverse the Möbius function). I will survey a sequence of generalisations of this principle: first to locally
finite posets (Hall and Rota) and to monoids with the finite-decomposition property (Cartier-Foata), then to Möbius categories (Leroux, Lawvere-Menni), and finally to the recent notion of decomposition space (joint work with Gálvez and Tonks), of which Waldhausen's S-construction is an important example (cf.~also Dyckerhoff-Kapranov).
References
I.Gálvez-Carrillo, J.Kock, A.Tonks.
Decomposition spaces, incidence algebras and M\"obius inversion I:
basic theory.
ArXiv:1512.07573.
I.Gálvez-Carrillo, J.Kock, A.Tonks.
Decomposition spaces, incidence algebras and M\"obius inversion II:
completeness, length filtration, and finiteness.
ArXiv:1512.07577.
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