  Given a span of spaces, one can form the homotopy pushout and then take the
homotopy pullback of the resulting cospan. We give a concrete description of
this pullback as the colimit of a sequence of approximations, using what we
call the zigzag construction. We also obtain a description of loop spaces of
homotopy pushouts. Using the zigzag construction, we reproduce generalisations
of the Blakers-Massey theorem and fundamental results from Bass-Serre theory.
We also describe the loop space of a wedge and show that it splits after
suspension. Our construction can be interpreted in a large class of
$(\infty,1)$-categories and in homotopy type theory, where it resolves the
long-standing open problem of showing that a pushout of 0-types is 1-truncated.
The zigzag construction is closely related to the James construction, but works
in greater generality.
