#
The bicategory of topological correspondences

##
Rohit Dilip Holkar

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
C*-correspondence H(X,\lambda) from
C^*(G,\alpha) to C^*(H,\beta). We
described the composition of two topological correspondences in one of
our earlier articles. In the present article, we prove that second
countable locally compact Hausdorff groupoids with Haar systems form
a bicategory T when equipped with topological
correspondences as 1-arrows and isomorphisms of topological
correspondences as 2-arrows.
On the other hand, it well-known that C*-algebras form
a bicategory C with C*-correspondences
as 1-arrows, and the unitary isomorphisms of Hilbert
C*-modules that intertwine the representations serve
as the 2-arrows. In this article, we show that a topological
correspondence going to a C*-one is a
bifunctor T to C. Finally, we show that in
the sub-bicategory of T consisting of the
Macho-Stadler-O'uchi correspondences, invertible 1-arrows are
exactly the groupoid equivalences.

Keywords:
Topological correspondences, bicategory of topological
correspondences, functoriality of topological correspondences

2020 MSC:
22D25, 22A22, 47L30, 46L08, 58B30, 46L89, 18D05.

*Theory and Applications of Categories,*
Vol. 38, 2022,
No. 22, pp 843-897.

Published 2022-07-05.

http://www.tac.mta.ca/tac/volumes/38/22/38-22.pdf

TAC Home