Let X be a quasi-projective complex variety. It follows from the work of Voevodsky that the motivic cohomology of X, denoted as H^p,q(X) where q and p are integers with q nonnegative, can be represented in the triangulated category of motives over the field of complex numbers, denoted as [special characters omitted]. That is, there exists an object [special characters omitted]_mot(q) in [special characters omitted] such that H^p,qX=Hom DM^eff,-NisM X,℘motq p-2q where M(X) is the motive of X. We construct objects [special characters omitted]_mor(q) and [special characters omitted]_Sing(q) in [special characters omitted] to represent the morphic cohomology L^qH ^p(X) and the singular cohomology [special characters omitted](X^an) of X. More precisely, L^qH^pX =HomDM^eff,-Nis MX,℘mor qp-2q H^pSingX^an =HomDM^eff,-Nis MX,℘Sing qp-2q where X is smooth. If X is singular, we define the morphic cohomology of X by the above formula. As an application, we show that Friedlander's comparison result L^qH^p(X) ≅ [special characters omitted](X^an), where X is smooth of pure dimension d and q ≥ d, can be generalized to singular varieties. As a second application, the morphic cohomology operations are considered.