Categorie Regolari

This thesis explores the theoretical foundations and properties of regular categories, a central concept in category theory with significant applications in various branches of mathematics. We begin by revisiting the definition of regular categories and establishing fundamental preliminaries. A category is termed regular if it possesses finite limits, stable regular epimorphisms under pullbacks, and coequalizers of kernel pairs. Chapter 1 provides a detailed analysis of epimorphisms and regular epimorphisms, including their definitions and properties within regular categories. We demonstrate that, in many algebraic contexts such as the categories of sets and abelian groups, regular epimorphisms coincide with surjective homomorphisms, thereby illustrating the intuitive nature of these morphisms in well-understood settings. In Chapter 2, we introduce exact sequences in regular categories and examine their role in the structural integrity of these categories. We provide rigorous proofs of key propositions, such as the preservation of regular epimorphisms under exact functors, and highlight the interplay between kernel pairs and coequalizers in defining exact sequences. Chapter 3 extends the discussion to equivalence relations and exact categories, presenting a categorical approach to relations and their equivalence properties. We delve into the implications of these equivalences for the structure and behavior of categories, drawing connections to classical set-theoretic equivalence relations and demonstrating their categorical analogs. Throughout the thesis, we emphasize examples and counterexamples from various categories, including sets, abelian groups, and modules over a ring, to illustrate the nuanced behaviors of regular categories.