Categorification

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위키데이터

• ID : Q5051825

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1. One diagram is worth a thousand words Each step of a “categorification process” should reveal more structure.
2. But categorifications are not unique, i.e. the category of finite-dimensional vector spaces can also be seen as a categorification of the natural numbers.
3. A topos (Lawvere) can be seen as a categorification of a Heyting algebra.
4. In mathematics, categorification is the process of replacing set-theoretic theorems with category-theoretic analogues.
5. Categorification, when done successfully, replaces sets with categories, functions with functors, and equations with natural isomorphisms of functors satisfying additional properties.
6. One form of categorification takes a structure described in terms of sets, and interprets the sets as isomorphism classes of objects in a category.
7. Using Khovanov’s categorification of the Weyl algebra, we investigate categorical structures arising from spherical harmonics.
8. Some folks are starting to talk more and more about “categorification”.
9. The following lists some common procedures that are known as categorification.
10. one could speak of G \mathbf{G} being “a categorification” of G G .
11. Some people also speak of horizontal categorification as categorification.
12. That is, the term ‘directed categorification’ works best for the combination of groupoidal categorification followed by laxification.
13. In the last sense of categorification, we start from a category in which certain equalities hold.
14. This mini-course will serve to introduce students to the new and exciting field of categorification.
15. Its goal is to prepare students for the workshop Geometric representation theory and categorification (part of the CRM thematic semester New Directions in Lie Theory).
16. The course will begin with a very brief review of the representation theory of associative algebras, before introducing the concept of weak categorification with some simple examples.
17. Categorification, a term coined by Louis Crane and Igor Frenkel, is the process of realizing mathematical structures as shadows of higher mathematics.
18. One reason for the prominence of quantum groups and Hecke algebras in categorification is that they provide a bridge between representation theory and low-dimensional topology.
19. That representation theory has proven to be an especially fertile ground for categorification is a fact that owes much to the geometric methods pervading the subject.
20. A particularly important object in geometric representation theory is the category of Soergel bimodules, which was used by Soergel to give a categorification of the Hecke algebra.
21. The term “categorification” was introduced by L. Crane and I. Frenkel to describe the process of realizing certain algebraic structures as shadows of richer higher ones.
22. In the past 15 years, it has become increasingly clear that categorification is actually a broad mathematical phenomenon with applications extending far beyond these original considerations.
23. Approaches to categorification vary but generally involve replacing set-theoretic statements by their category-theoretic analogues.
24. Representation theory provides an especially fertile ground for categorification.
25. "Categorification is an area of pure mathematics that attempts to uncover additional structure hidden in existing mathematical objects.
26. The term "categorification" was introduced about 15 years ago by Crane and Frenkel in an attempt to construct an example of 4-dimensional Topological Quantum Field Theory (TQFT for short).
27. The categorification of quantum groups is given by performing the Hall algebra construction we saw above for mathematical objects called sheaves.
28. This recovers the existing work on categorification of quantum groups by Khovanov, Lauda and Rouquier in a different language.
29. The main objective of the present proposal is to bring together mathematicians with international recognition whose research domains are related to categorification's problems.
30. Several conjectures on acyclic skew-symmetrizable cluster algebras are proven as direct consequences of their categorification via valued quivers.
31. The conference aims to illuminate current trends in categorification and higher representation theory and the diverse techniques that are being employed.
32. The following directions will be emphasized: -Techniques for categorification at roots of unity with associated applications to low-dimensional topology.
33. Categorification has led to many breakthroughs in representation theory in the last 15 years.
34. The PhD project will focus on both advancing the general theory and studying special classes of examples of categorification, e.g. coming from Soergel bimodules, which categorify Hecke algebras.
35. The term “categorification” was introduced by Louis Crane in 1995 and refers to the process of replacing set-theoretic notions by the corresponding category-theoretic analogues.
36. Upgrading the categorification to a p-dg algebra was done for quantum groups by Qi-Khovanov and Qi-Elias.
37. For some great examples of categorification see this list on MO, and for the meaning of categorification see this MO question or that article by Baez/Dolan.
38. The companion volume (Contemporary Mathematics, Volume 684) is devoted to categorification in geometry, topology, and physics.
39. In the last 20 years, major progress in representation theory, low-dimensional topology and related areas has been made through the process of categorification.
40. Cautis and Sussan conjectured a categorification of this correspondence within the framework of Khovanov's Heisenberg category.
41. I'll present recent advances in the categorification by foams of these structures and related knot invariants, before discussing open questions and conjectures.
42. Categorification is a powerful tool for relating various branches of mathematics and exploiting the commonalities between fields.
43. This volume focuses on the role categorification plays in geometry, topology, and physics.
44. The companion volume (Contemporary Mathematics, Volume 683) is devoted to categorification and higher representation theory.
45. We can offer limited funds to PhD students working on categorification in quantum topology, broadly defined, to attend the lectures in the first week.

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위키데이터

• ID : Q5051825