in many mathematical disciplines is one of the large goals of reaching a classification of the objects studied in the respective subrange. In many ranges also the modern research is still far from a complete classification, yet beginnings are to a partial classification one of the substantial sources of new terms and concepts.
Depending upon kind of the objects there are different definitions for it, which objects for the purposes of the classification are not to be regarded as “substantially differently” (isomorphic).
classification by enumerating
this kind of the classification exists in the indication of a complete list of the isomorphism classes. Examples are:
- Each vector space over a body k is isomorphic to k (n) for a certain cardinal number n.
- The classification of the finite simple groups.
See also: Standard format
classification by invariants
an invariant is a characteristic of an object, which is for all objects equal to an isomorphism class. A complete system of invariants is the indication of several characteristics, so that two objects, which agree in all these characteristics, are isomorphic. Examples are:
- Vector spaces over a firm body are clearly certain by the indication of their dimension up to isomorphism.
- Closed surfaces are clearly certain by the indication of their sex up to Diffeomorphie.
classification by equivalence of categories
a weak form of the classification is often reached by an equivalence from categories to a simpler category. Examples are:
- The category that divisor width run gene of a galoisschen body extension is equivalent for the category of the sub-groups of the Galoisgruppe.
- The category of the overlays of a topological area is equivalent under certain conditions for the category of the quantities with an operation of the Fundamentalgruppe of the basis area.