Compatible systems of $\mathrm{\ell }$-adic sheaves

We introduce a notion of compatibility for families $({\mathcal{F}}_{\mathrm{\ell }}{)}_{\mathrm{\ell }}$ of bounded constructible $\mathrm{\ell }$-adic complexes of 'etale sheaves on schemes. For schemes of finite type over a field, this notion is preserved by the usual six functors. We prove that the compatibility of a family is preserved by the nearby cycles functor and by the linearized $\epsilon$-factors introduced recently by the author. We establish independence of $\mathrm{\ell }$ for the characteristic cycles and characteristic $\epsilon$-cycles of compatible families.