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We develop the basic theory of nilpotent types and their localizations away from sets of numbers in Homotopy Type Theory. For this, general results about the classifying spaces of fibrations with fiber an Eilenberg–Mac Lane space are proven. We also construct fracture squares for localizations away from sets of numbers. All of our proofs are constructive.

Summary

This chapter approaches aspects of unstable homotopy theory from the perspective of the preceding chapter: We study the unstable analog of effective descent and thereby embark on an analysis of the algebro–geometric objects that arise from these considerations. This lends a modern perspective on a variety of classical topics: Hopf rings, the Ravenel–Wilson results on the infinite loopspaces associated to BP, Dieudonné theory, Brown–Gitler spectra, and the formal schemes associated to Eilenberg–Mac Lane spaces. We also include a novel discussion of the functorial algebraic geometry of Hopf rings, which gives a reinterpretation of the central Ravenel–Wilson relation and exerts an overall influence on our presentation of the other materials.

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Nilpotent types and fracture squares in homotopy type theory

4 - Unstable Cooperations

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2 results

Nilpotent types and fracture squares in homotopy type theory

4 - Unstable Cooperations

Summary