Given a mathematical structure, it is common that two substructures belong to the same isomorphism class. However, the way they are included in the whole structure can not be studied if they are identified. For example, in a finite-dimensional vector space, all subspaces of the same dimension are isomorphic, but must be distinguished to consider their intersection, sum, etc. The associative algebras consisting of coquaternions and 2 × 2 real matrices are isomorphic as rings. Yet they appear in different contexts for application (plane mapping and kinematics) 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.
Gi.ven a mathematical structure, it is common that two substructures belong to the same isomorphism class. However, the way they are included in the whole structure can not be studied if they are identified. For example, in a finite-dimensional vector space, all subspaces of the same dimension are isomorphic, but must be distinguished to consider their intersection, sum, etc The associative algebras consisting of coquaternions and 2 × 2 real matrices are isomorphic as rings. Yet they appear in different contexts for application (plane mapping and kinematics) 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.