Telescope of a commutative monoid

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 \[ \gamma _{M,x}\colon T(M,x)\to M^{\grp }. \]