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, 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, 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.

Related concepts

• Grothendieck group

• localization of a monoid

References

Classical accounts include

• Daniel Quillen, On the group completion of a simplicial monoid pdf, MIT preprint 1971, Memoirs of the AMS vol 529, 1994, pp. 89-105

• Peter May, $E_\infty$-Spaces, group completions, and permutative categories (pdf)

• William Dwyer, Daniel Kan, Simplicial localization of categories, Journal of pure and applied algebra 17 (1980) 267-284

$\infty$-Group completion is discussed in

• Ulrich Bunke, Georg Tamme, section 2.1 of Regulators and cycle maps in higher-dimensional differential algebraic K-theory (arXiv:1209.6451)

• Thomas NikolausAlgebraic 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 in

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