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. The term was coined by Louis Crane.[1][2]
The reverse of categorification is the 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. 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. They are used in a similar way to the words like 'generalization', and not like 'sheafification'.[4]
- ^ Crane, Louis; Frenkel, Igor B. (1994-10-01). "Four-dimensional topological quantum field theory, Hopf categories, and the canonical bases". Journal of Mathematical Physics. 35 (10): 5136–5154. arXiv:hep-th/9405183. doi:10.1063/1.530746. ISSN 0022-2488.
- ^ Crane, Louis (1995-11-01). "Clock and category: Is quantum gravity algebraic?". Journal of Mathematical Physics. 36 (11): 6180–6193. arXiv:gr-qc/9504038. doi:10.1063/1.531240. ISSN 0022-2488.
- ^ Khovanov, Mikhail; Mazorchuk, Volodymyr; Stroppel, Catharina (2009), "A brief review of abelian categorifications", Theory Appl. Categ., 22 (19): 479–508, arXiv:math.RT/0702746
- ^ Alex Hoffnung (2009-11-10). "What precisely Is "Categorification"?".
© MMXXIII Rich X Search. We shall prevail. All rights reserved. Rich X Search