高维代数
在数学中,特别是(高阶)范畴论中,高维代数是指对范畴化结构的研究。其在非阿贝尔代数拓扑与抽象代数的推广中有应用。
高维范畴
[编辑]定义高维代数的第一步是高阶范畴论中2-范畴的概念,以及二阶范畴的更“几何化”的概念。[1] [2][3]
更高级的概念因此定义为范畴的范畴,或称为超范畴。这将范畴的标记推广到高维——范畴被视为可以解释抽象范畴基本理论(ETAC)的劳维尔公理的任何结构。[4][5][6][7]
因此,超范畴可被视作元范畴、[8]多范畴、多图或有色图。 超范畴的概念于1970年被首次提出,[9]随后在理论物理(特别是量子场论和拓扑量子场论)、数理生物学及数理生物物理学中得到了应用。[10]
高维代数中的其他途径涉及:弱2-范畴、弱2-范畴的同态、可变范畴(又称索引或参数化范畴)、拓扑斯、增广范畴 以及内范畴。
二维广群
[编辑]高维代数中,二维广群是一维广群的推广,[11]后一种广群可视为所有态射都可逆的特殊范畴。
二维广群通常用来捕捉几何对象的信息,如高维流形(或n维流形)。[11]一般来说,一个n维流形是在局部上像是n维欧几里得空间的空间,而整体结构可能是非欧的。
1976年,罗纳德·布朗在ref.[11] 中首先提出了二维广群,并进一步发展了它在非阿贝尔代数拓扑中的应用。[12][13][14][15]与其相关的“双”概念指的是二维李代数胚,以及更一般的R代数体概念。
非阿贝尔代数拓扑
[编辑]应用
[编辑]理论物理
[编辑]在量子场论中有量子范畴[16][17][18]和量子二维广群。[18]我们可以把量子二维广群看作是通过2-函子定义的基本广群,这样就可由弱2-范畴Span(Groupoids)的视角思考量子基本广群(QFGs)这一物理上有意义的情况,然后为流形和配边构造2-希尔伯特空间和2-线性映射。下一步,我们将通过此类2-函子的自然变换来获得带角的配边。于是有说法称,在规范群SU(2)的作用下,“扩展的拓扑量子场论可以给出等同于量子引力的蓬扎诺-雷其模型的理论”;[18]相似地,图拉耶夫-维罗模型也可以通过SUq(2)的表示得到。因此,我们可以用对称性给出的变换广群来描述规范理论——或者许多种量子场论(QFTs)及局域量子物理的状态空间。例如,对于规范理论的情况,我们可以用作用于状态的度规变换来描述状态空间,在这种情况下状态就是连接。在与量子群相关的对称性的情况下,我们会得到量子广群的表示范畴(representation category)的结构,[16]而非广群的表示范畴的2-向量空间。
另见
[编辑]参考文献
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阅读更多
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