Given a spectrum, is there any kind of machinery that can tell you whether it is the Ktheory 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.

1$\begingroup$ If I recall correctly, Gunnar Carlsson posed a conjecture some time ago, asking if the Ktheory spectrum of a permutative category can be identified with the category of module spectra over a Ktheory spectra over a bipermutative category. Your question is a more generalized version of his question. $\endgroup$– user62675Commented Dec 8, 2014 at 1:43
1 Answer
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.