symmetric simplicial sets

One advantage of symmetric simplicial sets is that their model category structure is determined, cf. (*), by taking cofibrations as monomorphisms. It means that the classical homotopy category Ho (of CW-complexes) can be obtained just from the category S of symmetric simplicial sets (without any additional data). Simplicial sets do not have this property (it was also observed by J. H. Smith). In (*), we have used this property of S to give an elementary proof of an important result of D. Dugger saying that Ho is a free homotopy theory over a point. (*) J. Rosicky and W. Tholen, Left-determined model categories and universal homotopy theories, TAMS 355 (2003), 3611-3623.