"Categorification in mathematics"의 두 판 사이의 차이
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imported>Pythagoras0 (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...) |
imported>Pythagoras0 |
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* 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. | * 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). | * This approach (interpreted suitably) has been a remarkable success with the representation theory of semi-simple Lie algebras (and their associated quantum groups). | ||
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| + | ==related items== | ||
| + | * [[Categorification of quantum groups]] | ||
| + | * [[Monoidal categorifications of cluster algebras]] | ||
2015년 6월 14일 (일) 16:54 판
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).