# Is Margolis's axiomatisation conjecture still alive?

The construction of the category of finite spectra is easy, but there are different constructions of the whole homotopy category of spectra, all of which leading to the same result up to an equivalence. In his book, Spectra and the Steenrod Algebra, North-Holland, Amsterdam, N.Y., 1983, H.R. Margolis conjectured that there is a system of axioms, which characterizes the homotopy category in therm of finite spectra. If I do understand correctly, the results of M. Porta arXiv:0706.4458v2 and A. Heider arXiv:0707.0707v1 solve this problem for algebraic, respectively topological triangulated categories. I'm most familiar with algebraic triangulated categories, so it is very possible to occur misunderstanding in the topological case. But for the algebraic case it seems to me that the problem is settled: If $\mathcal T$ is an algebraic $\alpha$-well generated triangulated category, then it is equivalent to a fixed quotient of the derived category of the dg-algebra with several objects (these objects are the $\alpha$-compact objects of $\mathcal T$). Or, am I wrong?

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Margolis' conjecture is as follows: Let $\mathcal{S}$ be the stable homotopy category and $\mathcal{T}$ any compactly generated triangulated category. If the subcategories of compact objects are equivalent as triangulated categories $\mathcal{S}^c\simeq\mathcal{T}^c$ then $\mathcal{S}\simeq\mathcal{T}$. This conjecture is open. If $\mathcal{T}$ is known to be topological then it is a theorem in Schwede's Annals paper. The category $\mathcal{T}$ cannot be algebraic because $\mathcal{S}$ is not.
Thank you! Sure that Margolis's conjecture is formulated for homotopy category of spectra, but it may be easily generalized: If $\mathcal S$ and $\mathcal T$ are both $\alpha$-well generated, and ${\mathcal S}^\alpha\cong{\mathcal T}^\alpha$, is it true that ${\mathcal S}\cong{\mathcal T}$? After I posted the question, I realized: the property of being topological (or algebraic) is not expressed in terms of finite spectra (or, more generally $\alpha$-compact objects). –  George C. Modoi Jun 8 '11 at 12:09