An isomorphism class is a collection of mathematical object

Mathematical object

In mathematics and the philosophy of mathematics, a mathematical object is an abstract object arising in mathematics.Commonly encountered mathematical objects include numbers, permutations, partitions, matrices, sets, functions, and relations...

s 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

Ordinal number

In set theory, an ordinal number, or just ordinal, is the order type of a well-ordered set. They are usually identified with hereditarily transitive sets. Ordinals are an extension of the natural numbers different from integers and from cardinals...

 and graphs

Graph theory

In mathematics and computer science, graph theory is the study of graphs, mathematical structures used to model pairwise relations between objects from a certain collection. A "graph" in this context refers to a collection of vertices or 'nodes' and a collection of edges that connect pairs of...

. 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 coquaternion
  Coquaternion
  In abstract algebra, the split-quaternions or coquaternions are elements of a 4-dimensional associative algebra introduced by James Cockle in 1849 under the latter name. Like the quaternions introduced by Hamilton in 1843, they form a four dimensional real vector space equipped with a...
  
  s and 2 × 2 real matrices are isomorphic as rings
  Ring theory
  In abstract algebra, ring theory is the study of rings—algebraic structures in which addition and multiplication are defined and have similar properties to those familiar from the integers...
  
  . 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
  Fundamental group
  In mathematics, more specifically algebraic topology, the fundamental group is a group associated to any given pointed topological space that provides a way of determining when two paths, starting and ending at a fixed base point, can be continuously deformed into each other...
  
   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
  Abelian group
  In abstract algebra, an abelian group, also called a commutative group, is a group in which the result of applying the group operation to two group elements does not depend on their order . Abelian groups generalize the arithmetic of addition of integers...
  
   this isomorphism is non-unique. Furthermore, the classification of covering spaces makes strict reference to particular subgroups of , specifically distinguishing between isomorphic but conjugate
  Conjugacy class
  In mathematics, especially group theory, the elements of any group may be partitioned into conjugacy classes; members of the same conjugacy class share many properties, and study of conjugacy classes of non-abelian groups reveals many important features of their structure...
  
  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.