nLab group completion

Redirected from "group completions".
Contents

Not to be confused with completion of a group.

Context

Algebra

Group Theory

Monoid theory

Contents

Idea

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 1974, 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 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

References

Classical accounts:

and specifically concerning configuration spaces of points:

Discussion of \infty-group completion:

and specifically its monoidal properties:

Last revised on August 25, 2024 at 15:23:12. See the history of this page for a list of all contributions to it.