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Cyclic homology in a special world

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Abstract

In work of Connes and Consani, Γ-spaces have taken a new importance. Segal introduced Γ-spaces in order to study stable homotopy theory, but the new perspective makes it apparent that also information about the unstable structure should be retained. Hence, the question naturally presents itself: to what extent are the commonly used invariants available in this context? We offer a quick survey of (topological) cyclic homology and point out that the categorical construction is applicable also in an \({\mathbb N}\)-algebra (aka. semi-ring or rig) setup.

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Acknowledgements

Apart from obvious input from Alain Connes this tribute has benefitted from enlightening conversations with M. Brun, K. Consani, L. Hesselholt, M. Hill, and C. Schlichtkrull. Also, the preprint [8] by de Brito and Moerdijk and the papers of Santhanam [55] and Mandell [46] were inspirational. The author also wants to thank an anonymous referee for correcting an unfortunate misconception about events in the early 1980s.

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Dundas, B.I. (2019). Cyclic homology in a special world. In: Chamseddine, A., Consani, C., Higson, N., Khalkhali, M., Moscovici, H., Yu, G. (eds) Advances in Noncommutative Geometry. Springer, Cham. https://doi.org/10.1007/978-3-030-29597-4_5

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