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In mathematics, categorification is the process of replacing set-theoretic theorems by category-theoretic analogues. Categorification, when done successfully, replaces sets by categories, functions with functors, and equations by natural isomorphisms of functors satisfying additional properties. The term was coined by Louis Crane. Categorification is the reverse process of ''decategorification''. Decategorification is a systematic process by which isomorphic objects in a category are identified as equal. Whereas decategorification is a straightforward process, categorification is usually much less straightforward, and requires insight into individual situations. ==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. 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 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 much of the parallels in structure; for example 抄文引用元・出典: フリー百科事典『 ウィキペディア（Wikipedia）』 ■ウィキペディアで「categorification」の詳細全文を読む スポンサード リンク
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