I was giving a talk in a seminar, and I mistakenly said that the coskeleton tower of a quasi-category was its Postnikov tower. Someone corrected me, but a discussion then ensued about what, precisely, this tower is. It appears to be homotopy-invariant, and each $k$-coskeleton looks like it is somehow related to something along the lines of a weak $(k,1)$-homotopy-category for $k\geq 2$, but the other members of the seminar said that the $(k,1)$-homotopy-category is constructed differently and doesn't seem to be equivalent. What's interesting is this tower seems to converge to $X$ when $X$ is a quasicategory.
Has this tower been studied before? Does it have a name?