People often consider exceptional sets of objects (i.e. collections of objects satisfying certain strong orthogonality conditions: $Ext^{l}(P_i,P_j)$ should be zero for $l\neq 0$ + something else) in derived categories of coherent sheaves (over algebraic varieties; possibly the first example corresponds to the Beilnson's description of the derived category of coherent sheaves on the projective space of dimension $n$). Are there any examples of this notion in some stable homotopy categories (in the sense of abstract model categories; one can consider the category of modules over a ring spectrum here)?
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A natural topological example of a [higher] category with an exceptional collection is constructible complexes on a stratified space, locally constant along contractible strata. This holds with any coefficients  e.g. you can look at sheaves of Smodules or Emodules for an $E_\infty$ring spectrum if you prefer. The exceptionality encodes a. the contractibility of strata, b. the absence of Exts "in the wrong direction" for extensions of constant sheaves off the strata. I would presume you could also find examples of "derived Fano schemes" with exceptional collections  i.e. nontrivial analogs of the many (toric Fano e.g.) schemes admitting exceptional collections, but I don't know of such. 

