symmetric monoidal (∞,1)-category of spectra
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
$p$-completion is to p-adic homotopy theory as rationalization is to rational homotopy theory.
For more see at formal completion.
For $A$ an abelian group (or commutative ring) and $p$ a prime number, the $p$-completion of $A$ is the limit
(e.g. May Ponto, 10.1.1) For more see at formal completion.
$A$ is called $p$-complete if the canonical homomorphism $A \to A_p^\wedge$ is an isomorphism.
(…) (e.g. May-Ponto, 10.2)
The fracture theorem says that under mild conditions a (stable) homotopy type decomposes into its rationalization and its $p$-completions.
For $A = \mathbb{Z}$ the integers, the $p$-completion is the p-adic integers. (Notice that here traditionally one writes $\mathbb{Z}_p = \mathbb{Z}_p^\wedge$.)
More generally, if $A$ is finitely generated, then $A_p^\wedge \simeq A\otimes \mathbb{Z}_p$. (e.g. May Ponto, p. 154)
Classical accounts:
Dennis Sullivan, Geometric topology: localization, periodicity and Galois symmetry, volume 8 of K- Monographs in Mathematics. Springer, Dordrecht, 2005. The 1970 MIT notes, Edited and with a preface
by Andrew Ranicki (pdf)
Aldridge Bousfield, Daniel Kan, Localization and completion in homotopy theory, Bull. Amer. Math. Soc. 77 6 (1971) 1006-1010 [doi:10.1090/S0002-9904-1971-12837-9, pdf]
Aldridge Bousfield, Daniel Kan, Homotopy limits, completions and localizations, Lecture Notes in Mathematics, 304 Springer (1972) [doi:10.1007/978-3-540-38117-4]
Peter May, Kate Ponto, chapters 7 and 8 of More concise algebraic topology: Localization, completion, and model categories (pdf)
See also
Doug Ravenel, chapter 2, def. 2.1.14 Complex cobordism and stable homotopy groups of spheres
John Rognes, section 4.6 of The Adams spectral sequence, 2012 (pdf)
Last revised on January 22, 2023 at 20:38:03. See the history of this page for a list of all contributions to it.