Examples of categorification One form of categorification takes a structure described in terms of sets, and interprets the sets as isomorphism classes of objects in a category. For example, the set of natural numbers can be seen as the set of cardinalities of finite sets (and any two sets with the same cardinality are isomorphic). In this case, operations on the set of natural numbers, such as addition and multiplication, can be seen as carrying information about products and coproducts of the category of finite sets. Less abstractly, the idea here is that manipulating sets of actual objects, and taking coproducts (combining two sets in a union) or products (building arrays of things to keep track of large numbers of them) came first. Later, the concrete structure of sets was abstracted away - taken "only up to isomorphism", to produce the abstract theory of arithmetic. This is a "decategorification" - categorification reverses this step.
Other examples include homology theories in topology. Emmy Noether gave the modern formulation of homology as the rank of certain free abelian groups by categorifying the notion of a Betti number.[3] See also Khovanov homology as a knot invariant in knot theory.
An example in finite group theory is that the ring of symmetric functions is categorified by the category of representations of the symmetric group. The decategorification map sends the Specht module indexed by partition {\displaystyle \lambda }\lambda to the Schur function indexed by the same partition, (引用終り) 以上
