On the comparison of spans and bisets

Abstract: We compare the bicategory of spans with that of bisets (a.k.a. bimodules, distributors, profunctors) in the context of finite groupoids. We construct in particular a well-behaved pseudo-functor from spans to bisets. This yields an application to the axiomatic representation theory of finite groups, namely a new proof and a strengthening of Ganter and Nakaoka's identification of Bouc's category of biset functors as a reflective subcategory of global Mackey functors. To this end, we also prove a tensor-monadicity result for linear functor categories.