Groupoid $\Cst$-algebras have long provided useful and interesting examples of $\Cst$-algebras, and their role has increased in importance in recent years. In the context of the classification program, for example, a result of Li from 2020 shows that every classifiable $\Cst$-algebra is isomorphic to a twisted groupoid $\Cst$-algebra. In coarse geometry, one can construct a coarse groupoid for a metric space $X$ and use this structure in the study of the coarse Baum-Connes conjecture for $X$.

As such, it is of interest to investigate when inverse systems of groupoids dualize to direct limits of groupoid $\Cst$-algebras. Additionally, we were interested in writing $\sigma$-compact groupoids as limits of inverse systems of second-countable groupoids, in the hopes of extending some of the results which were only available in the second-countable setting.

If $\newcommand{\gpdG}{\mathcal{G}}\gpdG$ and $\newcommand{\gpdH}{\mathcal{H}}\gpdH$ are groupoids equipped with Haar measures, we will define what it means for a morphism $\gpdG \to \gpdH$ to be Haar system preserving, and show that such morphisms induce *-morphisms at the $\Cst$-algebra level. We will then apply this observation to inverse systems of groupoids. In particular, for a specific $\sigma$-compact groupoid $\gpdG$ equipped with a Haar system $\{\mu^x\}$ we will discuss how to construct an inverse system of second-countable groupoids whose limit is $\gpdG$. I will conclude the talk with a discussion of properties and applications of this construction.

This is based on joint work with Kyle Austin.