TITLE: Chromatic convergence of E(n)-localized homotopy categories
QUESTION [7 upvotes]: Given the Chromatic Convergence Theorem, can we state this globally as some convergence in the category of categories? That is, we have the subcategories of finite spectra in each stable homotopy category of $E(n)$-localized spectra, say $\mathcal{L}_n^{fin}$.  Is it correct that there should be "inclusion" functors $\mathcal{L}_n^{fin}\to\mathcal{L}_{n+1}^{fin}$? Can we say that if we take some kind of limit (maybe this should be phrased in terms of quasi-categories or something?) we get the homotopy category of finite spectra?

REPLY [8 votes]: As Fernando showed, you do have a functor from the category of finite spectra to an appropriate limit of the categories of $E(n)$-local finite spectra, and this recovers the mapping space between any pair of finite spectra:
$$
{\rm Map}(Y,X) \simeq {\rm holim}_n {\rm Map}(L_{E(n)} Y, L_{E(n)} X).
$$
However, this can't be made into an equivalence of categories because some non-finite objects show up as limits of finite objects.
For each $n$, choose a finite type $n$ complex $X_n$ whose $n$'th Morava $K$-theory is nontrivial (hence the $m$'th Morava $K$-theory is nontrivial for all $m \geq n$).  Being of type $n$ implies that $L_{E(k)} X_n \simeq *$ for $k < n$.  We therefore get a system of objects
$$
Y_n = L_{E(n)} (X_0 \vee X_1 \vee \dots \vee X_n)
$$
together with natural weak equivalences $L_{E(n-1)} Y_n \to Y_{n-1}$ induced by projecting off the factor $X_n$.  This assembles into an object in an appropriate homotopy limit category.
However, there is no finite complex $Y$ such that $L_{E(n)} Y \simeq Y_n$ for all $n$, because the number of cells necessary to build $Y_n$ grows in an unbounded fashion.  You can check this by observing that the $n$'th Morava $K$-theory of $Y_n$ has rank at least $n$ because each wedge factor contributes rank at least 1.