BCN Spring16 Workshop on Number Theory & K-theory
From zeta functions to Waldhausen's S-construction
Presenters
Joachim Kock
Abstract
The classical M\"obius 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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