Not to be confused with group completion.
Classical groups
Finite groups
Group schemes
Topological groups
Lie groups
Super-Lie groups
Higher groups
Cohomology and Extensions
Related concepts
(…)
(-complete group)
For a solid ring, a group is called -complete if it has a central series
(i.e. such that under commutator subgroups we have )
such that each quotient group (which is an abelian group by the previous property) carries the structure of an -module.
(completion of a group)
Given a solid ring and any group , there is a universal construction of an -complete group – the -completion of – given by the limit
over the codomain functor from the comma category of -complete groups under .
More generally:
(fiberwise -complete group)
A short exact sequence of groups
is fiberwise or relative -complete if is so (in the sense of Def. ).
(fiberwise completion of a group)
Given a solid ring and any short exact sequence of groups (1), there is a universal construction of a fiberwise -complete exact sequence
(profinite completion)
For (a cyclic group of prime order) and applied to finitely generated groups, -completion coincides with -profinite completion.
(Malcev completion)
For (the rational numbers) and applied to nilpotent groups, -completion coincides with Malcev completion.
(e.g. Bousfield & Kan 1972, p. 99)
Original articles:
Aldridge Bousfield, Daniel Kan, §2 of: 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, Chapter IV of: Homotopy limits, completions and localizations, Lecture Notes in Mathematics, 304 Springer (1972) [doi:10.1007/978-3-540-38117-4]
Further developments:
Last revised on July 16, 2022 at 21:21:09. See the history of this page for a list of all contributions to it.