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Regarding limits, can I suggest that the nlab page on limits should not start out talking about quasicategories? (-: I'm not clear exactly what your plan is, but the page you have called limits (domenicofiorenza) doesn't seem appropriate for an nlab page called "limit" instead of, say, "limit in a quasicategory". Perhaps I have misunderstood what you're doing.
I made a couple comments.
i don't have the energy to look into this right now, but am interested. If Mike finds a second I am sure you two will find a good notion of weighted (oo,1)-limit.
A W-weighted limit of would be an object Z such that for all Y. In particular, looking at on the left-hand side would give you a canonical set of maps from Z to X for all X, i.e. a map , but I don't see any reason for that map to be a bijection.
Yes, that seems right to me.
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