Cat1-objects in modified categories of interest

Normal subobjects and quotient objects in algebraic structures have an important place in terms of classification. For example, homomorphisms on a group can be categorized using isomorphism theorems, and if a group is simple or perfect, it can be expressed in terms of homomorphisms on this group. The central extensions of algebraic structures have also attracted considerable interest from mathematicians. Normal subobjects and quotient objects were used to achieve centralized extensions. Singular objects and commutator objects are useful objects for classifying central expansions. This thesis consists of five chapters. In the introduction, a short history was given about some of the studies in the literature on the topic to be studied. In the second chapter, some related structures and modified categories of interests are given in the literature together with their fundamental properties. In the third section, the notion of (pre)cat1-object in any modified category of interest and this object is linked to the crossed modules in . Then in the fourth chapter, singularity, commutator and central extensions were given in the modified categories of interest. In the fifth section, the crossed module versions of the notions given in this thesis were obtained as the application of the fourth section.