Spectral category is a category enriched in the symmetric monoidal category of spectra. Hence they are a nonlinear version of dg-categories. More precisely, pretriangulated idempotent complete spectral categories are a model of stable idempotent complete (infinity,1)-categories. This equivalence is recorded in
See also
Stefan Schwede, Enhancements of triangulated categories, pdf (broken link)
Stefan Schwede, Brooke Shipley, Stable model categories are categories of modules, Topology 42:1 (2003) 103–153, MR2003g:55034
Background material on symmetric spectra is in