The classical localization of topological spaces at a given set of primes $\mathcal{P}$ is a functor $\mathcal{T}\xrightarrow{(-)_{(\mathcal{P)}}}\mathcal{T}$ from a suitable category of topological spaces to itself that has seen many equivalent definition. It factors $\mathcal{T}\xrightarrow{(-)_{(\mathcal{P)}}}\mathcal{T}_{\mathcal{(P)}}\hookrightarrow\mathcal{T}$ through the full subcategory of $\mathcal{P}$-local members of $\mathcal{T}$ and is left adjoint to the inclusion. Thus is preserves all colimits. It also preserves some limits. It preserves homotopy pullbacks and in particular fibre sequences. Which brings me to my question.
What are the conditions for a homotopy limit to be preserved by localization? More importantly, how does one compute the localization of infinite homotopy limits?
As an illustration I wish to calculate the rational homotopy type of the component of the based mapping space $Map_*^{B\iota}(BU(1),BU(2))$ where $\iota:U(1)\hookrightarrow U(2)$ is the canonical inclusion. I write $BU(1)=\mathbb{C}P^{\infty}$ as the homotopy colimit over $\mathbb{C}P^n$. Then I have
$Map_*^{B\iota}(BU(1),BU(2))_\mathbb{Q}=\left(Map_*^{B\iota}(hocolim_n\mathbb{C}P^n,BU(2))\right)_\mathbb{Q}=\left(holim_n Map_*^{B\iota|}(\mathbb{C}P^n,BU(2))\right)_\mathbb{Q}$.
On the inside I have
$Map_*^{B\iota|}(\mathbb{C}P^n,BU(2))_\mathbb{Q}=Map_*^{B\iota|\mathbb{Q}}(\mathbb{C}P^n,BU(2)_\mathbb{Q})=Map_*^{B\iota|_\mathbb{Q}}(\mathbb{C}P^n,K(\mathbb{Q},2)\times K(\mathbb{Q},4))=K(\mathbb{Q},2)$
and naively commuting localization with the homotopy inverse limit i get
$Map_*^{B\iota}(BU(1),BU(2))_\mathbb{Q}=holim K(\mathbb{Q},2)=K(\mathbb{Q},2)$
What is the obstruction to performing this naive commutation?
