Etudiant : BRADAAI OUMAIMA

Filière : Master Mathématiques Pures (MMP)

Encadrant : Pr. CHOULLI HANAN

Annèe : 2024

Résumé : The theory of category was invented by the American mathematicians Samuel Eilenberg (1913-1998) and Saunders Mac Lane (1909-2005), and was developed by the French mathematician Alexander Grothendieck (1928-2014), indeed this theory would have not existed if it was not for the universe axiom and this last was introduced by Grothendieck. Category theory may be understood as a general theory of structure. The main idea of the category-theoretic approach is to describe the properties of structures in terms of morphisms between objects, instead of the description in terms of elements. Hence, set-theoretic notions of sets or spaces are replaced by objects, while elements are replaced by arrows or morphisms. Category theory may be viewed not as a generalisation of set theory, but as an alternative foundational language which allows to describe structure in a relative way, that is, defined in terms of relations with other structures. Therefore the structure of every object is specified by all morphisms between this object and other objects. The characteristic aspect of a category theory is that all constructions of this theory are provided in the language of diagrams, consisting of appropriate morphisms between given objects. In this sense, the concept of a mathematical structure as a set of elements equipped with some properties is not fundamental. The category-theoretic proofs are provided by showing the commutativity of diagrams, and usually involve such structural concepts as functors between categories, natural transformations between functors, as well as limits and adjunctions of functors, what has to be contrasted with structure-ignorant methods of set-theoretic formalism, based on proving the equality between the elements of sets. One of the first achievement of category theory certainly is the theory of abelian categories; they play an important role in homology and provides the correct frame for studying the problems of exact sequences. In particular, regular and exact categories are those which represent certain essential properties of exactness in abelian categories. Certainly, as examples, most categories are "Algebra-Like". Regular categories thus provide the correct framework for the development of relation theory, especially equivalence relations. In general terms, categorical algebra is algebra viewed and generalized from the perspective of category theory. Thus, it studies aspects of categorical constructions and category-type constructions that are in the spirit of pure algebra, this primarily includes the study of monoidal category theory and the corresponding internal notions of monoid objects, module objects, etc. In this report, we essentially introduce the notion of "regular category" which recaptures many "exactness properties" of abelian categories, but avoids requiring additivity, and which is very fundamental in category theory;