We define a bicategory with étale, locally compact groupoids as objects and suitable correspondences, that is, spaces with two commuting actions as arrows; the 2-arrows are injective, equivariant continuous maps. We prove that the usual recipe for composition makes this a bicategory, carefully treating also non-Hausdorff groupoids and correspondences. We extend the groupoid C * -algebra construction to a homomorphism from this bicategory to that of C * -algebra correspondences. We describe the C * -algebras of self-similar groups, higher-rank graphs, and discrete Conduché fibrations in our setup.
We prove that the forgetful functors from the categories of C * -and W * -algebras to Banach * -algebras, Banach algebras or Banach spaces are all monadic, answering a question of J.Rosický, and that the categories of unital (commutative) C * -algebras are not locally-isometry ℵ 0 -generated either as plain or as metric-enriched categories, answering a question of I. Di Liberti and Rosický.We also prove a number of negative presentability results for the category of von Neumann algebras: not only is that category not locally presentable, but in fact its only presentable objects are the two algebras of dimension ≤ 2. For the same reason, for a locally compact abelian group G the category of G-graded von Neumann algebras is not locally presentable.
This article continues the study of diagrams in the bicategory of étale groupoid correspondences. We prove that any such diagram has a groupoid model and that the groupoid model is a locally compact étale groupoid if the diagram is locally compact and proper. A key tool for this is the relative Stone-Čech compactification for spaces over a locally compact Hausdorff space.
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