论文标题
关系和道克综合体的Cosheaf表示
Cosheaf Representations of Relations and Dowker Complexes
论文作者
论文摘要
Dowker复合体是一种抽象的简单综合体，以直接的方式从二进制关系中构建。尽管有两种方法可以执行这种结构 - 复合物的顶点是代表关系的矩阵的行或列的列 - 这两个结构是同等的。本文表明，从关系中构建Dowker建筑群是一个非信仰的协变函数。此外，我们表明，可以通过将构造丰富成陶氏综合体的Cosheaf来忠实。 Cosheaf可以通过Dowker复合物上的整数重量函数来概括，这是该关系的完全同构不变。关系的cosheaf表示实际上体现了两个陶氏络合物，我们构建了一个交换两个复合物的双重函子。最后，我们探索了另一种Cosheaf，该Cosheaf检测了Dowker Complex本身成为忠实的函数的失败。
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.
