论文标题
定期简单组的最佳三角剖分
Optimal Triangulation of Regular Simplicial Sets
论文作者
论文摘要
Barratt神经（表示为$ b $）是一种函数，将简单的设置带到其非分类简单的poset神经上。有序的简单复杂$ bsd \，x $，即kan subdivision $ sd \，x $的barratt神经是对原始的简单集合$ x $的三角构造，因为有天然地图$ bsd $ bsd \，x \ x $ x $的几何学对某些同源性具有同性恋的实现。这是对任何简单集可以进行三角测量的结果的改进。 如果简单套件沿其$ n $ th的脸部嵌入，则据说简单套件是规律的。 $ bsd \，x \ to x $是$ x $的三角剖分，这是因为kan细分使简单套装定期，而$ bx $是$ x $的三角元，每当$ x $是常规的。在本文中，我们认为$ b $被解释为从常规到非单明性简单集的函子，不仅是任何三角剖分，而且实际上是最好的。我们的意思是从某种意义上说，$ b $是沿Yoneda嵌入的Barycentric细分的左KAN扩展。
The Barratt nerve, denoted $B$, is the endofunctor that takes a simplicial set to the nerve of the poset of its non-degenerate simplices. The ordered simplicial complex $BSd\, X$, namely the Barratt nerve of the Kan subdivision $Sd\, X$, is a triangulation of the original simplicial set $X$ in the sense that there is a natural map $BSd\, X\to X$ whose geometric realization is homotopic to some homeomorphism. This is a refinement to the result that any simplicial set can be triangulated. A simplicial set is said to be regular if each of its non-degenerate simplices is embedded along its $n$-th face. That $BSd\, X\to X$ is a triangulation of $X$ is a consequence of the fact that the Kan subdivision makes simplicial sets regular and that $BX$ is a triangulation of $X$ whenever $X$ is regular. In this paper, we argue that $B$, interpreted as a functor from regular to non-singular simplicial sets, is not just any triangulation, but in fact the best. We mean this in the sense that $B$ is the left Kan extension of barycentric subdivision along the Yoneda embedding.
