The bicategory of topological correspondences

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~$\mathfrak{T}\to\mathfrak{C}$.

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