p-adic homotopy theory (changes) in nLab

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Contents

Idea

In pp-adic homotopy theory one studies, for any prime number pp, simply connected homotopy types (of topological spaces, hence ∞-groupoids) all of whose homotopy groups have the structure of (finitely generated) modules over the p-adic integers ℤ p\mathbb{Z}_p – the pp-adic homotopy types. The central theorem (Mandell 01) says that the (∞,1)-category on pp-adic homotopy types is faithfully embedded into the (∞,1)-category of E-∞ algebras over an algebraically closed field of characteristic pp (Lurie, p. 70, cor. 3.5.15).

In this way pp-adic homotopy theory is directly analogous to rational homotopy theory with the rational numbers ℚ\mathbb{Q} replaced by the p-adic integers.

The fracture theorem says that under mild conditions, homotopy theory may be decomposed in a precise sense into rational homotopy theory and pp-adic homotopy theory for each prime pp.

• p-adic numbers

• p-completion, p-localization

• mod p Whitehead theorem

• p-adic geometry

• p-adic cohomology

• p-adic physics

• rational homotopy theory, real homotopy theory

References

• Michael Mandell, E ∞E_\infty-algebras and pp-Adic homotopy theory, Topology 40 (2001), no. 1, 43-94. (pdf)

  surveyed in: Algebraic models in pp-adic homotopy theory, YTM13, 2013 (pdf)

• Jacob Lurie, pp-Adic homotopy theory (pdf)

• Jacob Lurie, Rational and p-adic Homotopy Theory