nLab group completion


Not to be confused with completion of a group.



Group Theory

Monoid theory



The forgetful functor UU from abelian groups to commutative monoids has a left adjoint GG. 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:CMon (Grpd)AbGrp (Grpd) 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 NN is a topological monoid, let BNB N denotes its bar construction (“classifying space”) and ΩBN\Omega B N the loop space of that. Then this

NΩBN N \longrightarrow \Omega B N

represents the group completion of NN (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 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 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

and specifically its monoidal properties in

Last revised on October 26, 2023 at 11:49:01. See the history of this page for a list of all contributions to it.