Lemma 4.3.9. Let \(\PSp ^{(2)}(C)\) be the lax equalizer \[ \PSp ^{(2)}(C) := C_*^{\N }\times _{C_*^{\N }\times C_*^{\N }}\Ar (C_*^{\N }), \] where the map \(C_*^{\N }\to C_*^{\N }\times C_*^{\N }\) is \((\id ,\Omega ^2\sh )\) and the map from the arrow category is the source-target functor. Let \(\Sp ^{(2)}(C)\subseteq \PSp ^{(2)}(C)\) be the full subcategory spanned by those objects whose structure maps are isomorphisms. Then the even-level functor \[ E\colon \Sp (C) \to \Sp ^{(2)}(C), \qquad E(X)_k := X_{2k}, \] with structure maps \[ \widetilde {\sigma }^X_k\colon X_{2k} \xrightarrow {\sigma ^X_{2k}} \Omega X_{2k+1} \xrightarrow {\Omega (\sigma ^X_{2k+1})} \Omega ^2X_{2k+2}, \] is an equivalence.
Proof. For limits, let \(X\colon I \to \PSp (C)\) be a diagram and write \(\overline {X}\colon I \to C_*^{\N }\) for the underlying diagram of sequences. Since \(C_*\) has \(I\)-indexed limits and \(\Omega \) preserves limits, the endofunctor \(\Omega \sh \) preserves \(I\)-indexed limits. Hence the limit \(L := \lim _i \overline {X}_i\) in \(C_*^{\N }\) inherits a structure map \[ L \simeq \lim _i \overline {X}_i \to \lim _i \Omega \sh (\overline {X}_i) \simeq \Omega \sh (L), \] which makes it the limit in \(\PSp (C)\).
For colimits, let \(D := \colim _i \overline {X}_i\) in \(C_*^{\N }\). For every \(i\), the composite \[ \overline {X}_i \xrightarrow {\sigma ^{X_i}} \Omega \sh (\overline {X}_i) \to \Omega \sh (D) \] is compatible with the diagram, and therefore induces a structure map \(D \to \Omega \sh (D)\). We claim that the resulting prespectrum is the colimit. Indeed, for any prespectrum \(Y\) the hom anima \(\Hom _{\PSp (C)}(D,Y)\) is the equalizer of the two maps \[ \Hom _{C_*^{\N }}(D,Y) \rightrightarrows \Hom _{C_*^{\N }}(D,\Omega \sh Y) \] given by postcomposition with \(\sigma ^Y\) and precomposition with the structure map of \(D\), by Lemma 4.3.7. Since \(D\) is the colimit of the \(\overline {X}_i\) in \(C_*^{\N }\), and since limits commute with limits in \(\An \), this equalizer identifies with \[ \lim _i \Hom _{\PSp (C)}(X_i,Y), \] which is the desired universal property. □
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