Mgr. Josef Dvořák, Ph.D.

The main goal of the paper is a description of connected and projective objects of classes of categories that include categories of acts along with categories of pointed acts. In order to establish a general context and to unify the approach to both of the categories of acts, the notion of a concrete category with a unique decomposition of objects is introduced and studied. Although these categories are not extensive in general, it is proved in the paper that they satisfy a version of extensivity which ensures that every noninitial object is uniquely decomposable into indecomposable objects.

Let A and B be two abelian groups. The group A is called B-small if the covariant functor Hom(A,−) commutes with all direct sums B(k) and A is self-small provided it is A-small. The paper characterizes self-small products applying developed closure properties of the classes of relatively small groups. As a consequence, self-small products of finitely generated abelian groups are described.

The aim of the paper is to describe autocompact objects in Ab5-categories, i.e. objects in cocomplete abelian categories with exactness preserving filtered colimits, whose covariant Hom-functor commutes with copowers of the object itself. A characterization of non-auto compact object is given, a general criterion of autocompactness of an object via the structure of its endomorphism ring is presented and a criterion of autocompactness of products is proven.