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An algebraic representation of globular sets
Homology, Homotopy and Applications ( IF 0.5 ) Pub Date : 2020-01-01 , DOI: 10.4310/hha.2020.v22.n2.a8
Anibal M. Medina-Mardones 1
Affiliation  

We describe a fully faithful embedding of the category of (reflexive) globular sets into the category of counital cosymmetric $R$-coalgebras when $R$ is an integral domain. This embedding is a lift of the usual functor of $R$-chains and the extra structure consists of a derived form of cup coproduct. Additionally, we construct a functor from group-like counital cosymmetric \mbox{$R$-coalgebras} to $\omega$-categories and use it to connect two fundamental constructions associated to oriented simplices: Steenrod's cup-$i$ coproducts and Street's orientals. The first defines the square operations in the cohomology of spaces, the second, the nerve of higher-dimensional categories.

中文翻译:

球状集的代数表示

当 $R$ 是一个整数域时,我们描述了(自反)球状集合的类别完全忠实地嵌入到共同对称 $R$-coalgebras 的类别中。这种嵌入是对通常的 $R$ 链函子的提升,额外的结构由杯联积的派生形式组成。此外,我们构造了一个从类群共对称 \mbox{$R$-coalgebras} 到 $\omega$-categories 的函子,并使用它来连接与有向单形相关的两个基本构造:Steenrod's cup-$i$ coproducts 和 Street's东方人。第一个定义了空间上同调中的平方运算,第二个定义了高维范畴的神经。
更新日期:2020-01-01
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