JOUIWAL OF ALGEBRA 8, 77-95 (1968) Properties of Dense and Relative Adjoint Functors* FRIEDRICH ULMER Forschungsinstitut ft?r Mathematik der E.T.H., Ziirich, Switzerland and Mathematisches Institut der Universitat Heidelberg, Heidelberg, Germany Communicated by Saunders MaeLane Received October 10, 1966; revised December 11, 1966 INTRODUCTION In this paper we investigate some properties of dense1 and relative adjoint functors which we will use extensively in [I]. However it seems that some of these properties are of interest in themselves. Therefore we prefer not to include them in [Z] but to present them separately. Roughly speaking a functor J : M’ -+ M is dense (codense) if each object in M is canonically a direct (inverse) limit of objects jA4’, where M’ GM’. This is made precise in 1.3. Examples (cf. 1.5) are: M: the category of abelian groups, M’: the category of finitely generated abelian groups, J: inclusion; M: the category of rings, M’: the full subcategoryof M whoseonly object is the ring on two free generators,J: inclusion. Moreover if M’ is any category, then the Yoneda embedding YMr : M -+ (&TOpP, S), which assigns to M’ E M’ the contravariant horn-functor [-, W] from M’ to the category S of sets,is dense. The sameholds if M’ is additive and S is replaced by the category Ab.Gr. of abelian groups. The symbol (M’, Ab.Gr.) then denotes the category of additive functors. Denseinclusions 1: A’ -+ A are important because direct limit preserving functors on A are determined by their values on A’. For instance,let M’ be an abelian category with enough injectives. Denote by C*(M’, Ab.Gr.) the category of positive connected sequences (ri)jEz+ of functors M’ + Ab.Gr. (cf. [24 2.2). The universal property of the right satellitesS*t of an additive functor t : M’ --P Ab.Gr. implies that the functor S* : (M’, Ab.Gr.) -+ C*(M’, Ab.Gr.), t - S*t, is left adjoint to the forgetful functor C*(M’, Ab.Gr.) + (M’, Ab.Gr.), (rj)jez+ -+ fl. Hence S* is direct limit * Part of this work was supported by: Fonds ftir akademische NachwuchsfCrderung des Kantons Ziirich. IP. Gabriel informed me that he has also introduced this notion recently. He has independently observed 1.15. His results are to appear in [8]. 77