Isomorphism class

• The associative algebras consisting of coquaternions and 2 × 2 real matrices are isomorphic as rings. Yet they appear in different contexts for application so the isomorphism is insufficient to merge the concepts.
• In homotopy theory, the fundamental group of a space at a point, though technically denoted to emphasize the dependence on the base point, is often written lazily as simply if is path connected. The reason for this is that the existence of a path between two points allows one to identify loops at one with loops at the other; however, unless is abelian this isomorphism is non-unique. Furthermore, the classification of covering spaces makes strict reference to particular subgroups of, specifically distinguishing between isomorphic but conjugate subgroups, and therefore amalgamating the elements of an isomorphism class into a single featureless object seriously decreases the level of detail provided by the theory.