Categorification

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  1. One diagram is worth a thousand words Each step of a “categorification process” should reveal more structure.[1]
  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.[1]
  3. A topos (Lawvere) can be seen as a categorification of a Heyting algebra.[1]
  4. In mathematics, categorification is the process of replacing set-theoretic theorems with category-theoretic analogues.[2]
  5. Categorification, when done successfully, replaces sets with categories, functions with functors, and equations with natural isomorphisms of functors satisfying additional properties.[2]
  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.[2]
  7. Using Khovanov’s categorification of the Weyl algebra, we investigate categorical structures arising from spherical harmonics.[3]
  8. Some folks are starting to talk more and more about “categorification”.[4]
  9. The following lists some common procedures that are known as categorification.[5]
  10. one could speak of G \mathbf{G} being “a categorification” of G G .[5]
  11. Some people also speak of horizontal categorification as categorification.[5]
  12. That is, the term ‘directed categorification’ works best for the combination of groupoidal categorification followed by laxification.[5]
  13. In the last sense of categorification, we start from a category in which certain equalities hold.[6]
  14. This mini-course will serve to introduce students to the new and exciting field of categorification.[7]
  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).[7]
  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.[7]
  17. Categorification, a term coined by Louis Crane and Igor Frenkel, is the process of realizing mathematical structures as shadows of higher mathematics.[8]
  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.[8]
  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.[8]
  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.[8]
  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.[9]
  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.[9]
  23. Approaches to categorification vary but generally involve replacing set-theoretic statements by their category-theoretic analogues.[9]
  24. Representation theory provides an especially fertile ground for categorification.[9]
  25. "Categorification is an area of pure mathematics that attempts to uncover additional structure hidden in existing mathematical objects.[10]
  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).[10]
  27. The categorification of quantum groups is given by performing the Hall algebra construction we saw above for mathematical objects called sheaves.[10]
  28. This recovers the existing work on categorification of quantum groups by Khovanov, Lauda and Rouquier in a different language.[10]
  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.[11]
  30. Several conjectures on acyclic skew-symmetrizable cluster algebras are proven as direct consequences of their categorification via valued quivers.[12]
  31. The conference aims to illuminate current trends in categorification and higher representation theory and the diverse techniques that are being employed.[13]
  32. The following directions will be emphasized: -Techniques for categorification at roots of unity with associated applications to low-dimensional topology.[13]
  33. Categorification has led to many breakthroughs in representation theory in the last 15 years.[14]
  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.[14]
  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.[15]
  36. Upgrading the categorification to a p-dg algebra was done for quantum groups by Qi-Khovanov and Qi-Elias.[16]
  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.[17]
  38. The companion volume (Contemporary Mathematics, Volume 684) is devoted to categorification in geometry, topology, and physics.[18]
  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.[19]
  40. Cautis and Sussan conjectured a categorification of this correspondence within the framework of Khovanov's Heisenberg category.[20]
  41. I'll present recent advances in the categorification by foams of these structures and related knot invariants, before discussing open questions and conjectures.[20]
  42. Categorification is a powerful tool for relating various branches of mathematics and exploiting the commonalities between fields.[21]
  43. This volume focuses on the role categorification plays in geometry, topology, and physics.[21]
  44. The companion volume (Contemporary Mathematics, Volume 683) is devoted to categorification and higher representation theory.[21]
  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.[22]

소스

  1. ↑ 1.0 1.1 1.2 Categorification? A rough description
  2. ↑ 2.0 2.1 2.2 Categorification
  3. ↑ A Note on Categorification and Spherical Harmonics
  4. ↑ The n-Category Café
  5. ↑ 5.0 5.1 5.2 5.3 vertical categorification in nLab
  6. ↑ What precisely Is “Categorification”?
  7. ↑ 7.0 7.1 7.2 Introduction to Categorification
  8. ↑ 8.0 8.1 8.2 8.3 Categorification in Representation Theory
  9. ↑ 9.0 9.1 9.2 9.3 Categorification in Representation Theory
  10. ↑ 10.0 10.1 10.2 10.3 Categorification and Quantum Field Theories
  11. ↑ Categorification in algebraic geometry
  12. ↑ Some Consequences of Categorification
  13. ↑ 13.0 13.1 Categorification and Higher Representation Theory
  14. ↑ 14.0 14.1 CATEGORIFICATION IN REPRESENTATION THEORY (MIEMIETZV_U21SF)
  15. ↑ European Mathematical Society Publishing House
  16. ↑ Categorification of the Hecke algebra at roots of unity.
  17. ↑ Categorification of $\pi$?
  18. ↑ Categorification and higher representation theory
  19. ↑ Categorification in Representation Theory (MIEMIETZVU20SCIEP) at University of East Anglia on FindAPhD.com
  20. ↑ 20.0 20.1 Categorification in quantum topology and beyond
  21. ↑ 21.0 21.1 21.2 Categorification in Geometry, Topology, and Physics
  22. ↑ Activities

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