Mayer-Vietoris homotopy groups sequence of a pull-back of a fibration This must be an elementary question: could somebody tell me a reference for the Mayer-Vietoris homotopy groups sequence of a pull-back of a fibration?
I'm working in the category of pointed simplicial sets. So I've a pull-back of a (Kan) fibration of pointed simplicial sets, and I've read that in this situation you have an associated Mayer-Vietoris sequence relating the homotopy groups of the simplicial sets of the pull-back that looks like the classical Mayer-Vietoris sequence for the singular homology of a pair of open sets covering a topological space.
I've been searching in May's "Simplicial objects in Algebraic Topology" and Goerss-Jardine's "Simplicial Homotopy Theory", but I couldn't find it.
 A: A low level Mayer-Vietoris sequence for a pull-back of a fibration of groupoids is in 
(R. BROWN, P.R. HEATH and H. KAMPS), ``Groupoids and the
Mayer-Vietoris sequence'', {\em J. Pure Appl. Alg.} 30 (1983) 
and you will also find a version for coverings of groupoids in
`Topology and Groupoids', R. Brown (available on amazon.com). 
I've set as an exercise in my new coauthored book `Nonabelian algebraic topology' (see my web pages) to get a Mayer-Vietoris sequence for a pullback of a fibration of crossed complexes. 
A: Here is a lower-tech version of Reid's answer.  In general, if one has a Serre fibration p: E->B, and a map f: X->B, then there is a Mayer-Vietoris type sequence that comes from weaving together the long exact sequences in homotopy associated to p and to f*(p): f*(E)->X.  These exact sequences agree on every third term, because the fibers of f*(p) and p are homeomorphic.  The weaving process I'm talking about is an exercise in Hatcher's book (Exercise 38 in Section 2.2).
The relation between this exact sequence and the usual Mayer-Vietoris sequence in cohomology is explained here: Mathematically mature way to think about Mayer–Vietoris
A: I don't know of a reference, but here is a quick argument.  Suppose we want to compute the homotopy pullback P = X ×hZ Y of two maps f : X → Z and g : Y → Z of pointed simplicial sets.  Assume for convenience that everything is fibrant.  There is a fibration ZΔ[1] → Z∂Δ[1] = Z × Z with fiber ΩZ.  Now P is the pullback of the diagram X × Y → Z × Z ← ZΔ[1].  In particular, P → X × Y is also a fibration with fiber ΩZ, and the Mayer-Vietoris sequence follows from the long exact sequence of homotopy groups of this fibration.
