nLab group completion

Not to be confused with completion of a group.

Idea

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

$$K \;\colon\; CMon_\infty(\infty Grpd) \longrightarrow AbGrp_\infty(\infty Grpd)$$

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 1974, 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

$$N \longrightarrow \Omega B N$$

represents the group completion of $N$ (Quillen 71, section 9, May 1974, 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.

Examples

• group-completed configuration space of points

Related concepts

• Grothendieck group

• localization of a monoid

References

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)

• Peter May, §4 in: Infinite loop space theory, Bull. Amer. Math. Soc. 83 4 (1977) 456-494 [pdf, doi:10.1090/S0002-9904-1977-14318-8]

• 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]

• Sadok Kallel, §4.2.3 in: Configuration spaces of points: A user’s guide, Encyclopedia of Mathematical Physics 2nd ed, Elsevier (2024) [arXiv:2407.11092]

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:

• David Gepner, Moritz Groth, Thomas Nikolaus, Universality of multiplicative infinite loop space machines [arXiv:1305.4550]