In mathematics, categorification is the process of replacing set-theoretic theorems with category-theoretic analogues.
Categorification, when done successfully, replaces sets with categories, functions with functors, and equations with natural isomorphisms of functors satisfying additional properties.
Decategorification is a systematic process by which isomorphic objects in a category are identified as equal.
In the representation theory of Lie algebras, modules over specific algebras are the principal objects of study, and there are several frameworks for what a categorification of such a module should be, e.g., so called (weak) abelian categorifications.
[3] Categorification and decategorification are not precise mathematical procedures, but rather a class of possible analogues.
[4] One form of categorification takes a structure described in terms of sets, and interprets the sets as isomorphism classes of objects in a category.
In this case, operations on the set of natural numbers, such as addition and multiplication, can be seen as carrying information about coproducts and products 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.
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.
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
to the Schur function indexed by the same partition, essentially following the character map from a favorite basis of the associated Grothendieck group to a representation-theoretic favorite basis of the ring of symmetric functions.
This map reflects how the structures are similar; for example have the same decomposition numbers over their respective bases, both given by Littlewood–Richardson coefficients.