Recognition principle for connective spectra

Proof. Since \(\Omega ^{\infty }\) preserves finite products and \(\Sp \) is additive by Lemma 4.2.17, the fact that \(\Omega ^{\infty }\) uniquely lifts to \(\CGrp (\An )\) is an immediate consequence of the fact that \(\CGrp (-)\colon \Cat ^{\mathrm {prod}}_{\infty } \to \Cat _{\infty }^{\add }\) is right adjoint to the inclusion functor, see Proposition 5.3.23.

For the left adjoint, we use the equivalence \(\bB ^n\colon \CGrp (\An ) \iso \CGrp (\An _{\geq n})\) from Proposition 5.4.4, with inverse given by the iterated loop space functor \(\bOmega ^n\). For a commutative group \(G\), we may now define the spectrum \(\bB ^{\infty }G\) as follows: \[ \bB ^{\infty } G \quad := \quad (\bB ^n G, \sigma _n\colon \bB ^nG \iso \Omega \bB ^{n+1}G), \] where the structure maps \(\sigma _n\) are the units of the adjunction \(\bB \dashv \bOmega \). To see that this is a left adjoint to \(\bOmega ^{\infty }\colon \Sp \to \CGrp (\An )\), note that by additivity of \(\Sp \) we get an equivalence \(\Sp \simeq \CGrp (\Sp ) \simeq \Sp (\CGrp (\An ))\), so that we may write \(\Sp \) as the following limit: \[ \Sp \quad \simeq \quad \lim ( \, \cdots \xrightarrow {\Omega } \CGrp (\An ) \xrightarrow {\Omega } \CGrp (\An ) \xrightarrow {\Omega } \CGrp (\An )). \] We may then compute \begin {align*} \Hom _{\Sp }(\bB ^{\infty }G, X) &\simeq \lim _n \Hom _{\CGrp (\An )}(\bB ^n G, X_n) \\ &\simeq \lim _{n} \Hom _{\CGrp (\An )}(G, \Omega ^n X_n) \\ &\simeq \lim _{n} \Hom _{\CGrp (\An )}(G, X_0) \\ &\simeq \Hom _{\CGrp (\An )}(G, X_0). \end {align*}

Indeed, the structure isomorphisms \(\sigma _n\colon X_n \iso \Omega X_{n+1}\) induce compatible isomorphisms \(\Omega ^nX_n \iso \Omega ^{n+1}X_{n+1}\), identifying the resulting diagram constantly with \(X_0\). Since all these equivalences are natural in \(G\) and \(X\), this provides the adjunction. From this computation it also follows immediately that the unit \(G \to \bOmega ^{\infty }\bB ^{\infty }G\) of the adjunction is simply the identity map of \(G\), hence is an isomorphism, showing that the functor \(\bB ^{\infty }\colon \CGrp (\An ) \to \Sp \) is fully faithful. The essential image is the subcategory of the above limit given by \[ \CGrp (\An ) \quad \simeq \quad \lim ( \, \cdots \xrightarrow [\simeq ]{\Omega } \CGrp (\An _{\geq 2}) \xrightarrow [\simeq ]{\Omega } \CGrp (\An _{\geq 1}) \xrightarrow [\simeq ]{\Omega } \CGrp (\An )), \] which are precisely those spectra \(X\) such that \(X_n\) is \((n-1)\)-connected for all \(n \geq 1\), i.e. the connective spectra. □