Implementing the biset category of finite groups

We describe an implementation of the biset category of finite groups as a tower of standard categorical constructions, all of which are implemented in the software projec t CAP for algorithmic category theory. In particular, we describe the composition of bisets as a composition in a Kleisli category of some biadjunction monad. This composition relies on the universal property of the coequalizer completion of a group viewed as a groupoid on one object. Expressing this universal property offers an elegant categorical interpretation of the Schreier-Sims orbit algorithm. Indeed, the implementation relies on every aspect of the algorithm.