To each skeletally small abelian category one may associate a "spectrum" and a corresponding presheaf of localisations. The classical Zariski spectrum is the special case obtained from the category, mod-R, of finitely presented modules when R is a commutative noetherian ring. This fits into a general picture which links module theory, model theory, geometry and an additive version of topos theory via (anti-)equivalences between three 2-categories whose objects are, respectively, the small abelian categories, the locally coherent Grothendieck categories, the definable additive categories.