We propose a functorial concurrent semantics for Petri nets extended with read and inhibitor arcs, that we call inhibitor nets. Along the lines of the seminal work by Winskel on safe (ordinary) nets, the truly concurrent semantics is given at a categorical level via a chain of coreflections leading from the category SW-IN of semi-weighted inhibitor nets to the category Dom of finitary prime algebraic domains (equivalent to the category PES of prime event structures). As an intermediate semantic model, we introduce inhibitor event structures, an event based model able to faithfully capture the dependencies among events which arise in the presence of read and inhibitor arcs. Inhibitor event structures generalise several event structure models in the literature, like prime, asymmetric and bundle event structures.