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 -adic homotopy theory one studies, for any prime number , 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 – the -adic homotopy types. The central theorem (Mandell 01) says that the (∞,1)-category on -adic homotopy types is faithfully embedded into the (∞,1)-category of E-∞ algebras over an algebraically closed field of characteristic (Lurie, p. 70, cor. 3.5.15).
In this way -adic homotopy theory is directly analogous to rational homotopy theory with the rational numbers 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 -adic homotopy theory for each prime .
Michael Mandell, -algebras and -Adic homotopy theory, Topology 40 (2001), no. 1, 43-94. (pdf)
surveyed in: Algebraic models in -adic homotopy theory, YTM13, 2013 (pdf)
Jacob Lurie, -Adic homotopy theory (pdf)
Last revised on August 23, 2021 at 13:19:32. See the history of this page for a list of all contributions to it.