Delooping commutative groups

Proof. Both \(\bB \) and \(\bOmega \) preserve finite products: for \(\bB \) this is Lemma 5.1.16, while \(\bOmega \) preserves them because it is a right adjoint. We may therefore apply \(\CMon (-)\) to the adjunction.

For the first claim, it remains to argue that the forgetful functor \(\Mon (\An ) \to \An \) induces an equivalence \[ \CMon (\Mon (\An )) \iso \CMon (\An ). \] Since both \(\CMon (-)\) and \(\Mon (-)\) are defined as certain limit-preserving functors, there is an obvious equivalence \(\CMon (\Mon (\An )) \simeq \Mon (\CMon (\An ))\), and under this equivalence the previous functor corresponds to the forgetful functor \[ \Mon (\CMon (\An )) \to \CMon (\An ). \] But since \(\CMon (\An )\) is semiadditive by Proposition 5.3.20, this forgetful functor is an equivalence by Proposition 5.3.17.

Combining Lemma 5.4.3 with Remark 5.3.15, a commutative monoid \(X \in \CMon (\An )\) is a commutative group if and only if its monoid of path components \(\pi _0(X)\) is a group. We check that both functors land in \(\CGrp (\An )\): the anima \(\bB M\) is connected, so \(\pi _0(\bB M) = *\) is trivially a group; and for \(X \in \CMon (\An _*)\) we have \(\pi _0(\bOmega X) = \pi _1(X)\), which is a group because loop spaces are grouplike. Hence the adjunction restricts to commutative groups, and by Theorem 5.2.1 we get that it restricts to an equivalence

Here we use Remark 5.3.21 to pass between commutative monoids in pointed animae and pointed commutative monoids in animae. The claim for arbitrary \(n\) follows because \(\bB G\) is \(n\)-connected if and only if \(G \simeq \bOmega \bB G\) is \((n-1)\)-connected. □