Colimits of spectrum objects

Proof. The endofunctor \(\Omega \sh \) of \(C_*^{\N }\) induces an endofunctor, still denoted \(\Omega \sh \), of \(\PSp (C)\) by \[ (\Omega \sh X)_n := \Omega X_{n+1}, \] with structure maps \(\Omega (\sigma ^X_{n+1})\colon \Omega X_{n+1} \to \Omega ^2X_{n+2}\). The structure maps of \(X\) define a natural map \(\eta _X\colon X \to \Omega \sh X\). Since \(\sh \) preserves colimits and \(\Omega \) preserves sequential colimits, the endofunctor \(\Omega \sh \) preserves sequential colimits of prespectra.

For a prespectrum \(X\), define \[ X^{\mathrm {sp}} := \colim \left ( X \xrightarrow {\eta _X} \Omega \sh X \xrightarrow {\Omega \sh (\eta _X)} (\Omega \sh )^2X \to \cdots \right ) \] in \(\PSp (C)\). Then \[ \Omega \sh (X^{\mathrm {sp}}) \simeq \colim _{k \geq 0} (\Omega \sh )^{k+1}X \simeq \colim _{k \geq 0} (\Omega \sh )^kX = X^{\mathrm {sp}}, \] where the second isomorphism uses that the successor map \(k \mapsto k+1\) is cofinal in \(\N \). Under this isomorphism, the structure map of \(X^{\mathrm {sp}}\) is an isomorphism, so \(X^{\mathrm {sp}}\) is a spectrum.

It remains to prove the universal property. Let \(Y\) be a spectrum. Since \(\eta _Y\colon Y \to \Omega \sh Y\) is an isomorphism, composition with \(\eta _X\) induces an equivalence \[ \Hom _{\PSp (C)}(\Omega \sh X,Y) \xrightarrow {\sim } \Hom _{\PSp (C)}(X,Y); \] an inverse sends a map \(f\colon X \to Y\) to the composite \(\Omega \sh X \xrightarrow {\Omega \sh (f)} \Omega \sh Y \xrightarrow {\eta _Y^{-1}} Y\). Therefore \[ \Hom _{\Sp (C)}(X^{\mathrm {sp}},Y) \simeq \lim _k \Hom _{\PSp (C)}((\Omega \sh )^kX,Y) \simeq \Hom _{\PSp (C)}(X,Y), \] which is the desired adjunction. □