  We consider the homotopy category of complexes of projective modules over any
gentle algebra. We prove that indecomposable $\Sigma$-pure-injective objects in
s must be shifts of string or band complexes. We begin with a survey of purity
in compactly generated triangulated categories, recalling some
characterisations of $\Sigma$-pure-injective objects that mimic classical
results from the model theory of modules. We then specify to the aforementioned
homotopy category, and describe the compact objects in this case. Our proof
uses a recent adaptation of a classification technique known as the functorial
filtrations method. One of the key steps in our employment of this method is an
interpretation of the appropriate linear relations in terms of pp formulas in a
canonical multi-sorted language.
