By Corollary 5.4.5, every commutative monoid \(M\in \CMon (\An )\) admits a group completion \[ M \longrightarrow M^{\grp }=\bOmega \bB M. \] This is a universal construction, but in examples we often want a more explicit model for the underlying anima of \(M^{\grp }\).
For an ordinary commutative monoid \(A\) generated by an element \(x\), the group \(A^{\grp }\) admits a familiar description: it is obtained by formally inverting \(x\). Equivalently, one may take the sequential colimit \[ A \xrightarrow {x+(-)} A \xrightarrow {x+(-)} A \xrightarrow {x+(-)} \dots . \] The result is \(A[x^{-1}]\), which is already a group because \(x\) generates \(A\).
In animae the same construction is subtler. The path components behave as expected, but the higher homotopy can remember the symmetry coherences of the commutative monoid. The telescope theorem below gives a useful condition under which the same formal-inversion telescope computes the group completion.
Definition 5.5.1 (Telescope). Let \(M\in \CMon (\An )\) and let \(x\colon *\to M\) be a point. We define the telescope of \(M\) at \(x\) as the sequential colimit \[ T(M,x):=\colim \bigl (M \xrightarrow {x+(-)} M \xrightarrow {x+(-)} M \xrightarrow {x+(-)} \dots \bigr ) \qin \An . \]
Since the image of \(x\) is invertible in \(M^{\grp }\), the canonical map \(M\to M^{\grp }\) induces a comparison map \[ \gamma _{M,x}\colon T(M,x)\to M^{\grp }. \]
We use the following form of Nikolaus (2017), Proposition 6. Nikolaus gives several equivalent hypotheses; the abelian fundamental-group condition used here is the version needed for complex K-theory.
Theorem 5.5.2 (Group completion theorem, telescope form). Let \(M\in \CMon (\An )\) and let \(x\colon *\to M\) be a point. Assume that:
- (1)
-
The class \([x]\) generates the commutative monoid \(\pi _0(M)\);
- (2)
-
Each path component of \(T(M,x)\) has abelian fundamental group.
Then the comparison map \[ \gamma _{M,x}\colon T(M,x)\to M^{\grp } \] is an isomorphism of animae.
Proof. Under assumption (1), the singleton \(\{[x]\}\) is a generating set for \(\pi _0(M)\), and \(T(M,x)\) is the object denoted \(M_{\infty }\) by Nikolaus. The claim is therefore the implication (4) \(\Rightarrow \) (3) of Nikolaus (2017), Proposition 6. □
Remark 5.5.3. The abelian fundamental-group hypothesis is not merely cosmetic. Nikolaus explains that for the commutative monoid \(\bigsqcup _{n\geq 0}\bB \Sigma _n\), the telescope gives \(\Z \times \bB \Sigma _{\infty }\), while the group completion is obtained from this by plus construction. The full Proposition 6 in Nikolaus (2017) gives equivalent criteria in terms of a cyclic-permutation obstruction, and also a useful hypoabelian variant. We will only need the abelian case above.
Remark 5.5.4. Nikolaus also proves a homological group-completion theorem: after applying a multiplicative generalized homology theory, the map \(M\to \bOmega \bB M\) becomes algebraic localization under suitable Ore-type hypotheses. Its proof uses localization of ring spectra and their module categories. We will not develop that result in this book; the telescope criterion above is the part needed for complex K-theory.
Generated from the authoritative LaTeX source.