Classification (mathematics)
In many mathematical disciplines, one of the major goals is to classify the objects studied in the respective sub-area. In many areas, even modern research is still far from a complete classification , but approaches to a partial classification are one of the essential sources of new terms and concepts.
Depending on the type of objects, there are different definitions for which objects are to be regarded as "not substantially different" ( isomorphic ) for the purposes of classification .
Classification by enumeration
This type of classification consists in giving a complete list of isomorphism classes . Examples are:
- Every vector space over a field is isomorphic to for a certain cardinal number .
- The classification of finite simple groups .
Classification by invariants
An invariant is a property of an object that is the same for all objects of an isomorphism class . A complete system of invariants is the specification of several properties so that two objects that match in all of these properties are isomorphic. Examples are:
- Vector spaces over a solid body are uniquely determined by specifying their dimensions except for isomorphism .
- The main theorem about finitely generated Abelian groups classifies the finitely generated Abelian groups up to isomorphism.
- Closed areas are clearly defined by specifying their gender except for diffeomorphism .
Classification through equivalence of categories
A weak form of classification is often achieved through an equivalence of categories to a simpler category. Examples are:
- The category of the partial extensions of a Galois body extension is equivalent to the category of the subgroups of the Galois group .
- The category of superpositions of a topological space is under certain conditions equivalent to the category of sets with an operation of the fundamental group of the base space.