Given a spectrum, is there any kind of machinery that can tell you whether it is the K-theory spectrum of some recognizable category? For example, could TMF be realized in this way? In this case we would expect the chromatic height to decrease by one by the chromatic redshift conjecture. 



Is this spectrum the algebraic K-theory spectrum of something?
The standard algebraic $K$-theory constructions give connective spectra. So $TMF$ is not going to be in the image of such a construction. However Thomason showed that every connective spectrum can be constructed as the algebraic $K$-theory of some symmetric monoidal category (See Mandell's recent paper for a new proof of this result <http://arxiv.org/pdf/1002.3622.pdf>). This implies that $tmf$ is the algebraic $K$-theory of some symmetric monoidal category. Unfortunately, these inverse $K$-theory constructions do not appear to be amenable to chromatic analysis. 


