homotopy theory, (∞,1)-category theory, homotopy type theory
flavors: stable, equivariant, rational, p-adic, proper, geometric, cohesive, directed…
models: topological, simplicial, localic, …
see also algebraic topology
Introductions
Definitions
Paths and cylinders
Homotopy groups
Basic facts
Theorems
In $p$-adic homotopy theory one studies, for any prime number $p$, 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 $\mathbb{Z}_p$ – the $p$-adic homotopy types. The central theorem (Mandell 01) says that the (∞,1)-category on $p$-adic homotopy types is faithfully embedded into the (∞,1)-category of E-∞ algebras over an algebraically closed field of characteristic $p$ (Lurie, p. 70, cor. 3.5.15).
In this way $p$-adic homotopy theory is directly analogous to rational homotopy theory with the rational numbers $\mathbb{Q}$ 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 $p$-adic homotopy theory for each prime $p$.
Michael Mandell, $E_\infty$-algebras and $p$-Adic homotopy theory, Topology 40 (2001), no. 1, 43-94. (pdf)
surveyed in: Algebraic models in $p$-adic homotopy theory, YTM13, 2013 (pdf)
Jacob Lurie, $p$-Adic homotopy theory (pdf)