Representations of posets and tame matrix problems

Summary

§ 1. The theory of matrix problems can be considered in a natural way as a part or prolongation of homoiogical algebra (see [1]). Let K be an arbitrary category. Beside the notations Obj K, Mor K, Ind K and K(A,B) generally used for the collections of objects, of morphisms, of isoclasses of indecomposables and of morphisms from A to B(A, B ∈ Obj K), we shall also use the notation El K for the collection of all elements of all objects of K; this makes sense whenever K is a concrete category, i.e. a subcategory of the category of sets.

Let Φ be a functor with values in a concrete category V. To be precise, suppose that Φ depends on two variables belonging to categories K₁ and K₂ and that it is contravariant in the first and covariant in the second. Then El Φ will denote the collection of all elements of all sets Φ (A,B), where A ∈ Obj K₁, and B ∈ Obj K₂. With this rather strange terminology, we can say that one of the fundamental ideas of homological algebra consists in the remark that some categories G (of groups, algebras, modules…) can be investigated by means of a naturally constructed map h from El Φ to Obj K, where Φ happens to be one or another appearance of the functor Ext. The map h is surjective if the objects of G are considered up to isomorphisms, but it is far from being injective.