Not to be confused with completion of a group.
symmetric monoidal (∞,1)-category of spectra
Classical groups
Finite groups
Group schemes
Topological groups
Lie groups
Super-Lie groups
Higher groups
Cohomology and Extensions
Related concepts
monoid theory in algebra:
The forgetful functor $U$ from abelian groups to commutative monoids has a left adjoint $G$. This is called group completion. A standard presentation of the group completion is the Grothendieck group of a commutative monoid. As such group completion plays a central role in the definition of K-theory.
More generally in (∞,1)-category theory and higher algebra there is the left adjoint (∞,1)-functor
to the inclusion of abelian ∞-groups (connective spectra) into commutative ∞-monoids in ∞Grpd (E-∞ spaces). This may be called $\infty$-group completion.
This serves to define algebraic K-theory of symmetric monoidal (∞,1)-categories.
In algebraic topology a key role was played by models of this process at least on H-spaces (Quillen 71, May, def. 1.3). If $N$ is a topological monoid, let $B N$ denotes its bar construction (“classifying space”) and $\Omega B N$ the loop space of that. Then this
represents the group completion of $N$ (Quillen 71, section 9, May, theorem 1.6). This crucially enters the construction of the K-theory of a permutative category.
According to (Dwyer-Kan 80, prop. 3.7, prop. 9.2, remark 9.7) this indeed gives a model for the total derived functor of 1-categorical group completion.
Classical accounts:
Michael Barratt, Stewart Priddy, On the homology of non-connected monoids and their associated groups, Commentarii Mathematici Helvetici, 47 1 (1972) 1–14 [doi:10.1007/BF02566785, eudml:139496]
(on the Pontrjagin ring-structure under group completion of topological monoids)
Daniel Quillen: On the group completion of a simplicial monoid, Appendix Q in: Eric M. Friedlander, Barry Mazur: Filtrations on the homology of algebraic varieties, Memoirs of the AMS 529 110 (1994) 89-105 [doi:10.1090/memo/0529, pdf]
Peter May, $E_\infty$-Spaces, group completions, and permutative categories, in: New Developments in Topology, Cambridge University Press (1974) 61-94 [doi:10.1017/CBO9780511662607.008, pdf]
Dusa McDuff, Graeme Segal: Homology fibrations and the “group-completion” theorem, Inventiones mathematicae 31 (1976) 279-284 [doi:10.1007/BF01403148]
(on the Pontrjagin ring-structure under group completion of topological monoids)
William Dwyer, Daniel Kan, Simplicial localization of categories, Journal of pure and applied algebra 17 (1980) 267-284
and specifically concerning configuration spaces of points:
Graeme Segal, Configuration-spaces and iterated loop-spaces, Invent. Math. 21 (1973) 213-221 [doi:10.1007/BF01390197, pdf, MR 0331377]
Shingo Okuyama: The space of intervals in a Euclidean space, Algebr. Geom. Topol. 5 (2005) 1555-1572 [arXiv:math/0511645, doi:10.2140/agt.2005.5.1555]
Kazuhisa Shimakawa: Labeled configuration spaces and group completions, Forum Mathematicum (2007) 353-364 [doi:10.1515/FORUM.2007.014, pdf]
Discussion of $\infty$-group completion:
Ulrich Bunke, Georg Tamme, section 2.1 of Regulators and cycle maps in higher-dimensional differential algebraic K-theory (arXiv:1209.6451)
Thomas Nikolaus: Algebraic K-Theory of $\infty$-Operads (arXiv:1303.2198)
Ulrich Bunke, Thomas Nikolaus, Michael Völkl, def. 6.1 in Differential cohomology theories as sheaves of spectra (arXiv:1311.3188)
and specifically its monoidal properties:
Last revised on July 22, 2024 at 14:36:14. See the history of this page for a list of all contributions to it.