It is a theorem of Neil Strickland's that the category of harmonic spectra (i.e. the category of $p$-localized spectra localized at the infinite wedge of Morava K-theories) has no small objects. That is to say that there is not a single spectrum $X$ in the category of harmonic spectra such that $[X,\bigvee X\_i]=\bigoplus[X,X\_i]$. Take a finite spectrum in this category (for instance the sphere spectrum $\mathbb{S}$). Does anyone have an example of a coproduct in the category of harmonic spectra for which the above is NOT true? While Strickland's proof makes sense, it seems completely amazing to me. What HAPPENS to the sphere and the cells of finite spectra when we localize? It seems that if something were small to begin with, i.e. the above homotopy groups factored globally, killing some spectra shouldn't change this! Note that the same question can be asked of the $BP$-local category.


Are there conditions (other than finiteness) that we can put on the coproduct to make the statement true when the object on the left is finite (i.e. in the thick subcategory generated by the sphere)?



Counterexamples to Smallness of Harmonic Spectra
I believe that any example of a direct sum of harmonic spectra which isn't itself harmonic should answer the question. Namely, if the $X\_i, i \in \mathbb{N}$ are harmonic and $\bigoplus X\_i$ isn't, then the map 
$$ \bigoplus [S, X\_i] \to [S, L\_{\mathrm{harm}}\bigoplus X\_i]$$
isn't an isomorphism, because $\bigoplus X\_i \to L\_{\mathrm{harm}} (\bigoplus X\_i)$ isn't an equivalence. 


When the class of $E$-local spectra is closed under direct sums, then the $E$-local sphere is in fact compact and the $E$-localization functor commutes with (homotopy) colimits; then the localization functor is called smashing. (The $E\_n$-localization functors are smashing, as are the finite versions of them.) Conversely, if the $E$-local sphere is compact, then homotopy groups in the $E$-local category commute with direct sums, which means that direct sums in the $E$-local category are the same as direct sums in spectra (i.e., $E$-localization is smashing). 


So the point is that harmonic localization is not smashing. In fact, every finite spectrum (more generally, every suspension spectrum) is harmonic, and every spectrum is a homotopy colimit of finite spectra, so it suffices to produce a single example of a spectrum which isn't harmonic. An example is $H \mathbb{Z}/2$, which has no rational homotopy and no $K(n)$-homology for $n \geq 1$ (essentially because a formal group law can't have height $n$ and height $\infty$ at once). 


To be a little more explicit, you could find a directed system $X\_1 \to X\_2 \to \dots$ of finite (harmonic) spectra whose homotopy colimit is $H \mathbb{Z}/2$. Then the map $\bigoplus X\_i \to L\_{\mathrm{harm}} \bigoplus X\_i$ is not an equivalence (proof: the homotopy colimit of the directed system -- the mapping telescope -- is in the thick subcategory generated by the direct sum, and $\varinjlim X\_i $ is not harmonic). 


