Higher Groups in Homotopy Type Theory

Ulrik Buchholtz; Floris van Doorn; Egbert Rijke

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arXiv · 1802.04315

Higher Groups in Homotopy Type Theory

Ulrik Buchholtz·Floris van Doorn·Egbert Rijke

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Abstract

We present a development of the theory of higher groups, including infinity groups and connective spectra, in homotopy type theory. An infinity group is simply the loops in a pointed, connected type, where the group structure comes from the structure inherent in the identity types of Martin-Löf type theory. We investigate ordinary groups from this viewpoint, as well as higher dimensional groups and groups that can be delooped more than once. A major result is the stabilization theorem, which states that if an $n$-type can be delooped $n+2$ times, then it is an infinite loop type. Most of the results have been formalized in the Lean proof assistant.

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Ulrik Buchholtz, Floris van Doorn, Egbert Rijke. 2018-02-12. Higher Groups in Homotopy Type Theory. https://arxiv.org/abs/1802.04315

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