An Invitation to Categorification
An Invitation to Categorification
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DOI：
10.1090/noti2399
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发表时间：
2022-01
影响因子：
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通讯作者：
Aaron D. Lauda;Joshua Sussan
中科院分区：
文献类型：
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作者：
Aaron D. Lauda;Joshua Sussan
1.什么是分类？范畴化是将代数对象提升为具有更多结构的对象的过程。原始对象的所有结构都被保留下来，并且可以通过一个去范畴化过程来访问，该过程忘记了这个新的更高级别的结构。最初，这个术语表示用范畴概念取代基于集合的构造，其中等式被显式同构取代。然而，多年来，分类这个词已经被用在各种不同的上下文中，使得它的精确定义有些模糊。在大多数情况下，人们指定了一个严格的“非范畴化”过程，忘记了某种类型的结构。然后，范畴化是试图找到更复杂的结构的逆过程，该结构将非范畴化到某些特定的感兴趣对象。在这个大框架内有一些指导原则
1. What is Categorification? Categorification is the process of promoting an algebraic object to one with more structure. All of the structure of the original object is retained and can be accessed via a decategorification procedure that forgets this new higherlevel structure. Originally this term indicated the replacement of set-based constructions with categorical notions, where equalities are replaced by explicit isomorphisms. However, over the years, the term categorification has been used in a variety of different contexts, making its precise definition somewhat nebulous. In most cases, one specifies a rigorous “decategorification” procedure that forgets some type of structure. Categorification is then the inverse process of trying to find the more sophisticated structure that would decategorify to some specified object of interest. Within this broad framework are some guiding principles