Not to be confused with completion of a group.
symmetric monoidal (∞,1)-category of spectra
monoid theory in algebra:
The forgetful functor from abelian groups to commutative monoids has a left adjoint . 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 -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 is a topological monoid, let denotes its bar construction (“classifying space”) and the loop space of that. Then this
represents the group completion of (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, -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
-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 -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
Last revised on July 16, 2022 at 17:30:57. See the history of this page for a list of all contributions to it.