Higher Groups in Homotopy Type Theory

02/12/2018

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by   Ulrik Buchholtz, et al.

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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.