论文标题
拓扑通信的生物
The bicategory of topological correspondences
论文作者
论文摘要
众所周知，从局部紧凑的群体中，带有HAAR系统\（（（g，α）\）到另一个的拓扑对应\（（x，λ）\）产生a \（\ textrm {c}^*\） - \ sookence \ sosection \（\ coolccal cal cal cal）（\ cal）cal \（h，（h，β）\）。 \（\ textrm {c}^*（g，α）\）to \（\ textrm {c}^*（h，β）\）。在我们之前的一篇文章中，我们描述了两个拓扑通讯。在本文中，我们证明，使用HAAR系统的第二个可计数局部紧凑的Hausdorff拓扑组形成了一个bicateGory \（\ mathfrak {t} \），当时配备了拓扑相应的对应关系为1箭头。保存度量家族的拓扑对应的同态同态是〜\（\ mathfrak {t} \）中的2箭头。另一方面，众所周知，\（\ textrm {c}^*\） - 代数形成一个bicateogry \（\ mathfrak {c} \），带有\（\ textrm {c}^*\） - socumentence。 \（\ mathfrak {c} \）中的2个arrows是希尔伯特\（\ textrm {c}^*\）的单位 - 模块，使表示表示。在本文中，我们表明，拓扑通信将用于\（\ textrm {c}^*\） - 一个是双functor〜\（\ mathfrak {t} \ to \ mathfrak {c} \）。
It is known that a topological correspondence \((X,λ)\) from a locally compact groupoid with a Haar system \((G,α)\) to another one, \((H,β)\), produces a \(\textrm{C}^*\)-correspondence \(\mathcal{H}(X,λ)\) from \(\textrm{C}^*(G,α)\) to \(\textrm{C}^*(H,β)\). In one of our earlier article we described composition two topological correspondences. In the present article, we prove that second countable locally compact Hausdorff topological groupoids with Haar systems form a bicategory \(\mathfrak{T}\) when equipped with a topological correspondences as 1-arrows. The equivariant homeomorphisms of topological correspondences preserving the families of measures are the 2-arrows in~\(\mathfrak{T}\). One the other hand, it well-known that \(\textrm{C}^*\)-algebras form a bicateogry \(\mathfrak{C}\) with \(\textrm{C}^*\)-correspondences as 1-arrows. The 2-arrows in \(\mathfrak{C}\) are unitaries of Hilbert \(\textrm{C}^*\)-modules that intertwine the representations. In this article, we show that a topological correspondence going to a \(\textrm{C}^*\)-one is a bifunctor~\(\mathfrak{T}\to\mathfrak{C}\).
