• Homological Stability
• Algebraic Topology
• Homology of groups

We do not know many examples of sequences of groups fitting the categorical framework of pre-braided homogeneous categories, but for which we don't have connectivity, and homological stability fails. Peter Patzt constructed such an example with symmetric groups, and the goal of this thesis is to present other examples of the same flavour. Chapter one provides all the algebraic tools needed later, as well as the necessary theory about homology of groups. Chapter two introduces homological stability in the categorical framework of pre-braided locally homogeneous categories. For pairs of objects in such categories we construct the associated sequence of spaces, and formulate the connectivity axiom. We give also a proof of an homological stability theorem (only for integral homology), which states that the connectivity implies stability for the sequence of automorphism groups associated to the pair. The last chapter presents a nice criteria for checking 0-connectivity, and examples of unstable sequences of groups fitting into pre-braided homogeneous categories. We present examples in symmetric groups, general linear groups and projective general linear groups.