In this paper we consider a construction in an arbitrary triangulated category TT which resembles the notion of a Moore spectrum in algebraic topology. Namely, given a compact object C of TT satisfying some finite tilting assumptions, we obtain a functor which “approximates” objects from the module category of the endomorphism algebra of C in TT . This provides a higher analogue of a construction of Jørgensen which appears in (Manuscr Math 110:381–406, 2003) in connection with lifts of certain homological functors of derived categories. We show that this new functor is well-behaved with respect to short exact sequences and distinguished triangles, and as a consequence we obtain a new way of embedding a module category in a triangulated category. As an example of the theory, we recover Keller’s canonical embedding of the module category of a path algebra of a quiver with no oriented cycles into its u-cluster category of u⩾2u⩾2 .