A Family of Acyclic Functors

(1) find colim-acyclic objects in Ab. Here, Ab denote the category of abelian groups and Ab denote the (abelian) functor category for the small category C. The functor colim : Ab → Ab is the direct limit functor and F ∈ Ab is colim-acyclic if colimi F = 0 for i ≥ 1 (see [16] and [5], and the classical books of Cartan and Eilenberg [2] and of MacLane [13]). It is clear that if F is projective then it is colim-acyclic but, in the same way as not every flat module is projective (see, for example, [16, Section 3.2]), we may be missing colim-acyclic objects if we just consider projective ones. We shall assume the hypothesis that the category C is a graded partially ordered set (a graded poset for short). These are special posets in which we can assign an integer to each object (called the degree of the object) in such a way that preceding elements are assigned integers which differs in 1. Thus a graded poset can be divided into a set of “layers” (the objects of a fixed degree), and these layers are linearly ordered. Any simplicial complex (viewed as the poset of its simplices with the inclusions among them) and any subdivision category is a graded poset. Also, every CW -complex is (strong) homotopy equivalent to a simplicial complex, and thus to a graded poset. To attack problem (1) we start giving a characterization of the projective objects in Ab . Recall that for any small category C (not necessarily a poset), the projective objects in Ab are well known to be, by the Yoneda Lemma, summands of direct sums of representable functors. Moreover, if C is a poset with the descending chain condition (not necessarily graded) then [3, Corollary 3] the projective objects in Ab are also direct sums of representable functors (see also [11, Proposition 7] and [4, Theorem 9] for related results). In case C is a graded poset we characterize the projective functors in Ab as those functors which satisfy two conditions: