Isomorphism class

An isomorphism class is a collection of mathematical objects isomorphic to each other.

Isomorphism classes are often defined if 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 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 $X$ at a point $p$, though technically denoted $\pi_1(X,p)$ to emphasize the dependence on the base point, is often written lazily as simply $\pi_1(X)$ if $X$ 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 $\pi_1(X,p)$ is abelian this isomorphism is non-unique. Furthermore, the classification of covering spaces makes strict reference to particular subgroups of $\pi_1(X,p)$, 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.