关系的Cosheaf表示与Dowker复形
Cosheaf Representations of Relations and Dowker Complexes
摘要：
Dowker复形是一种抽象的单纯复形，它是以一种简单的方式从二元关系构造的。虽然有两种方法来执行这种构造-复形的顶点是表示关系的矩阵的行或列-这两种构造是同伦等价的。本文证明了从一个关系构造Dowker复形是一个不忠实的共变函子。此外，我们表明，这个函子可以通过丰富的建设成为一个cosheaf Dowker复杂的忠实。上层可以由Dowker复形上的整数权函数来概括，该权函数是该关系的完全同构不变量。一个关系的cosheaf表示实际上包含了两个Dowker复形，我们构造了一个对偶函子来交换这两个复形。最后，我们探讨了一个不同的cosheaf检测失败的Dowker复杂本身是一个忠实函子。
英文摘要：
The Dowker complex is an abstract simplicial complex that is constructed from a binary relation in a straightforward way. Although there are two ways to perform this construction -- vertices for the complex are either the rows or the columns of the matrix representing the relation -- the two constructions are homotopy equivalent. This article shows that the construction of a Dowker complex from a relation is a non-faithful covariant functor. Furthermore, we show that this functor can be made faithful by enriching the construction into a cosheaf on the Dowker complex. The cosheaf can be summarized by an integer weight function on the Dowker complex that is a complete isomorphism invariant for the relation. The cosheaf representation of a relation actually embodies both Dowker complexes, and we construct a duality functor that exchanges the two complexes. Finally, we explore a different cosheaf that detects the failure of the Dowker complex itself to be a faithful functor.
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