HDA的分支神经与Kan条件
The branching nerve of HDA and the Kan condition
摘要：
人们可以将任何严格的球状$ω$ -第三类增强的单纯神经（称为球状神经、分支和合并的半立方体神经）关联起来。如果这个严格球形$ω$ -范畴是由一个预三次集自由生成的，那么相应的同调论就包含了关于由预三次集模拟的高维自动机的几何的不同信息。在这个$ω$ -范畴中，对维数大于2的任何态射和关于维数大于1的任何合成律添加逆并不改变这些同调论。在这样的框架下，球状神经总是满足Kan条件。另一方面，分支和合并神经都不能满足它，除非在一些非常特殊和无趣的情况下。本文引入了两条新的神经（分支和合并半球神经），它们满足Kan条件，并分别与分支和合并半球神经具有相同的单纯同调.后者猜想是有关的薄元素猜想已经介绍了我们以前的文件。
英文摘要：
One can associate to any strict globular $ω$-category three augmented simplicial nerves called the globular nerve, the branching and the merging semi-cubical nerves. If this strict globular $ω$-category is freely generated by a precubical set, then the corresponding homology theories contain different informations about the geometry of the higher dimensional automaton modeled by the precubical set. Adding inverses in this $ω$-category to any morphism of dimension greater than 2 and with respect to any composition laws of dimension greater than 1 does not change these homology theories. In such a framework, the globular nerve always satisfies the Kan condition. On the other hand, both branching and merging nerves never satisfy it, except in some very particular and uninteresting situations. In this paper, we introduce two new nerves (the branching and merging semi-globular nerves) satisfying the Kan condition and having conjecturally the same simplicial homology as the branching and merging semi-cubical nerves respectively in such framework. The latter conjecture is related to the thin elements conjecture already introduced in our previous papers.
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