Basic question about the definition of the homology of a spectrum

Question

The general definition of the homology of a spectrum $E$ with coefficients in an abelian group $G$ is $$H_*(E;G):=\pi_*(E\wedge HG)$$ and I always see people using the equality $$H_*(E;G)=\mathrm{colim}_nH_{*+n}(E(n);G)$$ and say that it is easily seen to be the same. I tried to write things down but I can see why these two things coincide. Is it not obvious or have I missed something ?

Answer 1

Every spectrum $\{E_n\}$ is the same as $\operatorname{hocolim} \Sigma ^{-k} \Sigma ^\infty E_k$. If we have a spectrum $F$ and we want $E$'s homology with respect to $F$ we do as you say and consider $\pi_*(E \wedge F)$. By the above remark, this is the same as $\pi_*(\operatorname{hocolim} \Sigma ^{-k} \Sigma ^\infty E_k)$. So we have that $F_*(E)=\pi_*(E \wedge F)=\pi_*(\operatorname{hocolim} \Sigma ^{-k} \Sigma ^\infty E_k \wedge F)$.

Now, morally, since smashing is left adjoint to taking function spectra it commutes with homotopy colimits (I can't find a reference for this immediately. It should follow from left quillen functors preserving homotopy colimts, but you can work it out by hand in this case using the handicraft definition of smash product here), so we have one further equality $\pi_*(\operatorname{hocolim} \Sigma ^{-k} \Sigma ^\infty E_k \wedge F)=\pi_*(\operatorname{hocolim}(\Sigma^{-k} (\Sigma^\infty E_k \wedge F)))$. Now $\pi_*$ commutes with directed homotopy colimits in spectra for the same reason it does in spaces, the sphere spectrum is built out of finite spaces. This means we have the equality $\pi_*(\operatorname{hocolim}(\Sigma^{-k} (\Sigma^\infty E_k \wedge F)))=\operatorname{colim} \pi_*(\Sigma^{-k} (\Sigma^\infty E_k \wedge F))= \operatorname{colim} F_*(E_k)$.