The bicategory of topological correspondences

Abstract: It is known that a topological correspondence ((X,\lambda)) from a locally compact groupoid with a Haar system ((G,\alpha)) to another one, ((H,\beta)), produces a (\textrm{C}^*)-correspondence (\mathcal{H}(X,\lambda)) from (\textrm{C}^*(G,\alpha)) to (\textrm{C}^*(H,\beta)). 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}).