  As a generalization of DHR analysis, the superselection sectors are studied
in the case of absence of the spectrum condition for the reference
representation. Considered a net of local observables in the 4-dimensional
Minkowski spacetime, we show that it is possible to associate to a set of
representations, that are local excitations of a reference one fulfilling Haag
duality, a symmetric tensor $\mathrm{C}^*$-category $\Bim$ of bimodules of the
net, with subobjects and direct sums. The existence of conjugates is studied
introducing an equivalent formulation of the theory in terms of the presheaf
associated with the observable net. This allows us to find, under the
assumption that the local algebras in the reference representation are properly
infinite, necessary and sufficient conditions for the existence of conjugates.
Moreover, we present several results that suggest how the mentioned assumption
on the reference representation can be considered essential also in the case of
theories in curved spacetimes.
