The spectrum of the real line

Motivated by the study of persistence modules over the real line, we investigate the category of linear representations of a totally ordered set. We show that this category is locally coherent and we classify the indecomposable injective objects up to isomorphism. These classes form the spectrum, which we show to be homeomorphic to an ordered space. Moreover, as the spectral category turns out to be discrete, the spectrum parametrises all injective objects. Finally, for the case of the real line we show that this topology refines the topology induced by the interleaving distance, which is known from persistence homology.