TITLE: Exceptional collections of objects in topological triangulated categories?
QUESTION [6 upvotes]: 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)?

REPLY [4 votes]: 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 S-modules or E-modules 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.