A categorification of cycle class maps consists to define
realization functors from constructible motivic sheaves to other
categories of coefficients (e.g. constructible $l$-adic sheaves), which are compatible with the six operations. Given a field $k$, we
will describe a systematic construction, which associates,
to any cohomology theory $E$, represented in $DM(k)$, a
triangulated category of constructible $E$-modules $D(X,E)$, for $X$
of finite type over $k$, endowed with a realization functor from
the triangulated category of constructible motivic sheaves over $X$.
In the case $E$ is either algebraic de Rham cohomology (with $char(k)=0$), or $E$ is $l$-adic cohomology, one recovers in this way the triangulated categories of $D$-modules or of $l$-adic sheaves. In the case $E$ is rigid cohomology (with $char(k)=p>0$), this construction provides a nice system of $p$-adic coefficients which is closed under the six operations.