Implementing the biset category of finite groups

TL;DR

This paper details an implementation of the biset category of finite groups within the CAP software, utilizing categorical constructions and providing a categorical interpretation of the Schreier-Sims orbit algorithm.

Contribution

It introduces a categorical implementation of the biset category in CAP, connecting it with universal properties and the Schreier-Sims algorithm.

Findings

Implementation of biset composition as a Kleisli category composition.

Categorical interpretation of the Schreier-Sims orbit algorithm.

Integration of categorical constructions in computational group theory.

Abstract

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.