@Gustavo, right you are, but we (or at least I) don't have access to Lemma 2 in the source: the question has two answers, depending on that. Of course, rectification of this first step to a Postnikov tower is where the real math lies either way.

The analogy with topology is too close for this to be anything but canonical. I have nothing against Ronnie's choice, but it is not in common use. The most recent paper about Thomason's theorem and its generalizations is available on my web page as math.uchicago.edu/~may/PAPERS/MSZ.pdf, which gives background and references. No apologies needed. A good opportunity to advertise a classical observation that still seems much less well-known than it should be.

I agree with David, and it should be clear that the simplicial construction works, using the standard simplicial filtration.