Isomorphism class

In mathematics, an isomorphism class is a collection of mathematical objects which are isomorphic to each other.^[1]

Isomorphism classes are considered to specify that the difference between two mathematical objects is considered irrelevant.

Definition in category theory

Isomorphisms and isomorphism classes can be formalized in great generality using the language of category theory. Let ${\displaystyle C}$ be a category. A morphism ${\displaystyle f:A\to B}$ is called an isomorphism if there is a morphism ${\displaystyle g:B\to A}$ such that ${\displaystyle gf={\text{id}}_{A}}$ and ${\displaystyle fg={\text{id}}_{B}}$. Consider the equivalence relation that regards two objects as related if there is an isomorphism between them. The equivalence classes of this equivalence relation are called isomorphism classes.

Examples

Examples of isomorphism classes are plentiful in mathematics.

• Two sets are isomorphic if there is a bijection between them. The isomorphism class of a finite set can be identified with the non-negative integer representing the number of elements it contains.
• The isomorphism class of a finite-dimensional vector space can be identified with the non-negative integer representing its dimension.
• The classification of finite simple groups enumerates the isomorphism classes of all finite simple groups.
• The classification of closed surfaces enumerates the isomorphism classes of all connected closed surfaces.
• Ordinals are essentially defined as isomorphism classes of well-ordered sets (though there are technical issues involved).

However, there are circumstances in which the isomorphism class of an object conceals vital information about it.

• 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.^[opinion]
• In homotopy theory, the fundamental group of a space ${\displaystyle X}$ at a point ${\displaystyle p}$, though technically denoted ${\displaystyle \pi _{1}(X,p)}$ to emphasize the dependence on the base point, is often written lazily as simply ${\displaystyle \pi _{1}(X)}$ if ${\displaystyle X}$ 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 ${\displaystyle \pi _{1}(X,p)}$ is abelian this isomorphism is non-unique. Furthermore, the classification of covering spaces makes strict reference to particular subgroups of ${\displaystyle \pi _{1}(X,p)}$, 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.

References