TITLE: Homotopical perspective on the long exact sequence in homology and Mayer-Vietoris
QUESTION [7 upvotes]: For a while I have been wondering whether the long exact sequence in homology and the Mayer-Vietoris sequence can be phrased in homotopical terms. Recently, I heard that both may be reformulated via homotopy pullbacks/pushouts.
I am looking for a reference that develops this viewpoint, or at least mentions it in detail. I have not studied enough algebraic topology to fill in the blanks myself, nor have I worked enough with homotopy (co)limits to feel comfortable with them.
I have seen a couple of MSE answers which poke what I am looking for, but I cannot complete the picture myself.

REPLY [2 votes]: I haven't seen the Mayer–Vietoris sequence in the answers here, but it can be seen as a special example of the cofiber sequence in the accepted answer. 
Given a pointed excisive triad $(X;U,V)$, replace $X$ with the double mapping cylinder (homotopy pushout) $X'$ of the inclusions of $W = U ∩ V​$ in $U$ and $V$, with the interval through the basepoint collapsed to a point. Then the cofiber sequence starts
$$U \vee V \longrightarrow X' \longrightarrow \Sigma W \longrightarrow \Sigma U \vee \Sigma V.$$
It's a nice exercise to check the last map can be seen as the difference of the suspensions of the inclusions from $W$ to $U$ and $V$. Applying $H^*$ (or some other cohomology theory), the first two maps turn into ring maps (so products of coboundaries are zero), and the last one a map of graded groups (actually, of $H^*X$-modules). Its desuspension is the standard "difference of restrictions" map $H^*U \times H^*V \to H^* W$.