Telescope group completion theorem

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.