Categorification in mathematics
imported>Pythagoras0님의 2015년 6월 14일 (일) 17:50 판 (새 문서: ==introduction== * general motivation for categorification * algebraic/geometric structures <-> category * we can use general properties of the category $\mathcal{C}$ * It's a long es...)
introduction
- general motivation for categorification
- algebraic/geometric structures <-> category
- we can use general properties of the category $\mathcal{C}$
- It's a long established principle that an interesting way to think about numbers as the sizes of sets or dimensions of vector spaces, or better yet, the Euler characteristic of complexes.
- You can't have a map between numbers, but you can have one between sets or vector spaces.
- For example, Euler characteristic of topological spaces is not functorial, but homology is.
- One can try to extend this idea to a bigger stage, by, say, taking a vector space, and trying to make a category by defining morphisms between its vectors.
- This approach (interpreted suitably) has been a remarkable success with the representation theory of semi-simple Lie algebras (and their associated quantum groups).