分解空间与限制种
Decomposition spaces and restriction species
摘要：
我们证明了Schmitt的限制类（如图，拟阵，偏序集等）。自然诱导分解空间（naturally induced decomposition spaces）单位2-Segal空间），并且它们的相关余代数是分解空间的关联余代数的一般构造的一个实例。我们引入了有向限制种的概念，它继承了Schmitt的限制种，并导出了分解空间。普通的限制种是有限集和有限内射范畴上的预层，而有向限制种是有限偏序集和有限凸映射范畴上的预层。我们还介绍了monoidal（定向）限制种的概念，这导致monoidal分解空间，因此双代数，最常见的霍普夫代数。这个概念的例子包括有根森林、有向图、偏序集、双偏序集和许多相关的结构。由此产生的关联双代数的一个突出例子是有根树的Butcher-Connes-Kreimer霍普夫代数。普通和定向限制物种被证明是一个例子的分解空间的建设，从某些cocarrhythmfibrations的有限序数的范畴，也carrhythmover凸映射。证明依赖于一些美丽的单纯组合学，其中凸性的概念起着关键作用。开发的方法是独立的兴趣作为技术构造分解空间。
英文摘要：
We show that Schmitt's restriction species (such as graphs, matroids, posets, etc.) naturally induce decomposition spaces (a.k.a. unital 2-Segal spaces), and that their associated coalgebras are an instance of the general construction of incidence coalgebras of decomposition spaces. We introduce the notion of directed restriction species that subsume Schmitt's restriction species and also induce decomposition spaces. Whereas ordinary restriction species are presheaves on the category of finite sets and injections, directed restriction species are presheaves on the category of finite posets and convex maps. We also introduce the notion of monoidal (directed) restriction species, which induce monoidal decomposition spaces and hence bialgebras, most often Hopf algebras. Examples of this notion include rooted forests, directed graphs, posets, double posets, and many related structures. A prominent instance of a resulting incidence bialgebra is the Butcher-Connes-Kreimer Hopf algebra of rooted trees. Both ordinary and directed restriction species are shown to be examples of a construction of decomposition spaces from certain cocartesian fibrations over the category of finite ordinals that are also cartesian over convex maps. The proofs rely on some beautiful simplicial combinatorics, where the notion of convexity plays a key role. The methods developed are of independent interest as techniques for constructing decomposition spaces.
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