The Burnside bicategory of groupoids (Q2398330)

This paper considers actions of groupoids on sets. If \(G\) is a groupoid, then a functor \(X:G\to\mathbf{Set}\) is called a (left) \(G\)-set. the author constructs the \textit{Burnside bicategory of groupoids}, denoted by \(\mathbf{B}\), and shows that it has a morphism from the category \(\mathbf{G}\) of groupoids, functors, and natural transformations, called \textit{stabilization} and a compatible functor from \(\mathbf{G}_{cf}^{\mathrm{op}}\), where \(\mathbf{G}_{cf}\) is the sub-bicategory generated by finite covers, which provides \textit{transfers}. An \(H\times G\) bi-set is a functor \(X:H^{\mathrm{op}}\times G\to\mathbf{Set}\), while a \textit{correspondence} of groupoids (also called a \textit{span}) is a certain diagram of the form \(H \leftarrow K\to G\), where the functor \(K\to H\) satisfies suitable conditions. The author constructs the bicategory \(\mathbf{C}\) of correspondences, and proves that it is related to the bicategory \(\mathbf{B}\) by means of certain co-end and translation category constructions.