It seems to me that there should be some not too nebulous motivation for such a question. Rather than answer the question as stated, I'll give an open problem along the same lines. It starts with an old and much neglected paper: Daniel M. Kan Semisimplicial spectra Illinois J. Math. Volume 7 (1963), 463-478. That gives an analogue of based simplicial sets that allows for simplices of negative definition and is designed to give an alternative definition of spectra. There are problems with the smash product and there are several later papers that flesh out the theory (Kan and Whitehead, Burghelea and coauthors). I would be curious to see how this definition fits into the modern world of spectra. There is a notion of Kan semisimplicial spectrum (opus cit) and it seems very natural to wonder if there is an interesting version of stable quasicategories sitting as a subcategory of the category of Kan semisimplicial spectra.

It seems to me that there should be some not too nebulous motivation for such a question. Rather than answer the question as stated, I'll give an open problem along the same lines. It starts with an old and much neglected paper: Daniel M. Kan Semisimplicial spectra Illinois J. Math. Volume 7 (1963), 463-478. That gives an analogue of based simplicial sets that allows for simplices of negative dimension and is designed to give an alternative definition of spectra. There are problems with the smash product and there are several later papers that flesh out the theory (Kan and Whitehead, Burghelea and coauthors). I would be curious to see how this definition fits into the modern world of spectra. There is a notion of Kan semisimplicial spectrum (opus cit) and it seems very natural to wonder if there is an interesting version of stable quasicategories sitting as a subcategory of the category of Kan semisimplicial spectra.