Roman, it was Thomason, not Quillen who used it. Jonathan, I've answered your question, if only with an advertisement for a book I'm writing. Patricia, I like your answer. There is no really good reference yet, to my mind, since there is much more of interest to be said than is in any published reference. As an aside, I'd like to put in a good word for your first reference, Babson and Kozlov: that explains when you can hope that the nerve functor on categories with G-actions commutes with colimits, such as passage to orbits. That came up in recent work on equivariant classifying spaces.

Sorry to blow my own horn, but I am teaching things related to this in the REU I run at University of Chicago, and I'm writing a book that will include an exposition of subdivisions of categories and some neat combinatorial relationships (due to students, not me) between that and other notions, certainly including the factorization you mention (which is probably the best definition of the subdivision of a category). Note that as a composite of left and right adjoints, this is not a categorically well-behaved construction.

The quick answer to ``Is there a nice relation'' is no. For example, if $n\geq 3$ (if my memory is correct), $CX$ doesn't know the difference between $X = \Delta[n]$ and $X = \pa\DE[n]$.

Technical care is needed here: $G$-CW complex has a precise meaning, just like CW complexes but cells of the form $G/H \times D^n$'. So when $G$ is compact Lie, $n$-skeleta do not have geometric dimension $n$. Subcomplex must be taken in this equivariant sense, a union of equivariant cells. Then if $X^H \to Y^H$` is a weak homotopy equivalence, $X\to Y$ is the inclusion of a SDR. All bets are off for other guesses as to what a $G$-CW complex means, and for orbits replacing fixed points, and for merely nonequivariant SDR's. Just not in the cards.

This is Example 4.1.3 in Ravenel's ``Complex cobordism and stable homotopy groups of spheres'', where it is quite simply explained.

That is actually on my `to do' list. I hope to get to it someday. A hopefully student friendly introduction, but with perhaps too much focus on our own approach for your purposes, is ``Modern foundations for stable homotopy theory'', by Elmendorf, Kriz, Mandell, and myself (In Handbook of algebraic topology and on my web page). It is easier going than our book.

Just thanks to you Peter; I read the question and was thinking of writing an answer, and then I saw that you had already written the answer I might have written (except that I would have been too lazy to give exact references).

Naive Spanier-Whitehead works fine equivariantly and gives the full subcategory of finite G-CW spectra (up to equivalence) in any other good construction. Neil's answer is fine, with R the regular representation, but Gaunce's result I cited is valid for orbits of compact Lie groups, not just finite ones. The pre-EKMM Lewis-May foundations work for compact Lie groups and any universe, the suspension G-spectra of spheres with trivial $G$-action are cofibrant there, and space level maps can be used via adjunction. Passing to EKMM doesn't clarify things here.