Isomorphism classes are often defined as the exact identity of the elements of the set is considered irrelevant, and the properties of the structure of the mathematical object are studied. Examples of this are
ordinals and
graphs. However, there are circumstances in which the isomorphism class of an object conceals vital internal information about it; consider these examples:
The
associative algebras consisting of
coquaternions and 2 × 2
realmatrices 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.[opinion]
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.