The loop-space functor in homological algebra

This note constitutes a sequel to [AC], the terminology and notation of which are used throughout. Its purpose is to contribute some technical devices, viz. the notion of an ideal, and that of the loop-space functor, to the study of homological algebra. The concept of an ideal of an additive category is introduced (?1) in analogy with the familiar notion of the theory of rings. In particular the quotient of a category by a two-sided ideal is defined. Of especial interest are the ideals generated by identity maps, which are studied in ?2. For an abelian category with enough projectives a functor Q, defined on the quotient of the category by its projectives (?3), bears a close analogy to the loop-space functor of topology. This functor, as an operation on modules, has also been considered by P. J. Hilton and B. Eckmann [H-E]. If 3e is an abelian category with enough projectives then for 'C8, the category of proper s.e.s. in 3X, a functor r is introduced (?4). Pursuing the topological analogy, r plays the role of the operation which turns inclusion maps into fibre maps. This is related to the loop-space functor by the operation (?5) that its threefold iterate is just Q for the category 3C' (a fact which is known for the topological analogue). Using this result it is possible to associate to a proper s.e.s. exact sequences of homomorphism groups in the original category modulo its projectives. Finally the Ext groups of the original category are related to the homomorphism groups in the quotient by projectives (?6) by a functor which exhibits the latter as quotient groups of the former. The kernel measures the extent to which projectives fail to be injective; if they are all injective the study of Ext is reduced to that of the quotient category and the functor U. As in [AC] the theory is subject to straightforward dualization which however is barely indicated. The general reference is [AC]. To this should be added [G] which had not yet appeared when [AC] was written. It should be noted that "additive category" in [G] is used in a sense different from the present one, and that "abelian category" in [G] is equivalent to the "exact category" of Buchsbaum and [AC]. 1. Ideals in an additive category. A left ideal i of an additive category 3C is a subset of the maps of 3C with the properties