"Categorification in mathematics"의 두 판 사이의 차이
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imported>Pythagoras0 |
Pythagoras0 (토론 | 기여) |
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| 2번째 줄: | 2번째 줄: | ||
* general motivation for categorification | * general motivation for categorification | ||
* algebraic/geometric structures <-> category | * algebraic/geometric structures <-> category | ||
| − | * we can use general properties of the category | + | * we can use general properties of the category <math>\mathcal{C}</math> |
* 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. | * 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. | * You can't have a map between numbers, but you can have one between sets or vector spaces. | ||
2020년 11월 13일 (금) 17:40 기준 최신판
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).
articles
- Matsuoka, Takuo. “A Generalization of Categorification, and Higher ‘Theory’ of Algebras.” arXiv:1509.01582 [math], September 7, 2015. http://arxiv.org/abs/1509.01582.