The aim of this chapter is mainly to introduce some basic categorical concepts dealing with General Algebra, and to illustrate in this context many important properties of one of the most remarkable categories in this area, namely the category Grp of groups.

These concepts are basic in the sense that they deal with notions as elementary as, for instance, those of epimorphism, monomorphism, kernel, cokernel and equivalence relation.

The first section is devoted to setting basic facts concerning the internal equivalence relations in a given finitely complete category.

In the second section different notions of epimorphisms are considered and classified, and their stability properties are studied. We prove a new result concerning strong epimorphisms, which turns out to be a powerful tool in many situations. We call this result the Barr-Kock Theorem, in reference to an important and weaker version of it given in [1]. The notions of regular and Barr-exact categories are then recalled, and their main basic properties are proved. In this context the previous distinctions of different kinds of epimorphisms are pretty well simplified.

In Algebra, among the several notions of monomorphism (or, equivalently, of subobject), the class of normal subgroups in the category Grp of groups, or the class of ideals in the category Rng of rings naturally deserve great attention. The third section deals with the general notion of normal monomorphism. It introduces the concept of protomodular category, where this notion of normal monomorphism becomes intrinsic, precisely as this happens in Grp or Rng.