Corollary 5.4.5 (Group completion). The inclusion \(\CGrp (\An ) \hookrightarrow \CMon (\An )\) admits a left adjoint \[ (-)^{\grp }\colon \CMon (\An ) \to \CGrp (\An ) \] called group completion. It is given on objects by \(M \mapsto \bOmega \bB M\).
Proof. It follows immediately from the proposition that for every commutative group \(G \in \CGrp (\An )\), precomposition with the unit \(M \to \bOmega \bB M\) defines an equivalence \[ \Hom _{\CGrp (\An )}(\bOmega \bB M, G) \iso \Hom _{\CMon (\An )}(M,G), \] since both sides are equivalent to \(\Hom _{\CMon (\An _*)}(\bB M, \bB G)\). □
Generated from the authoritative LaTeX source.