THE TODD-COXETER PROCEDURE AND LEFT KAN EXTENSIONS

We introduce a generalization of the Todd-Coxeter procedure for the en umeration of cosets. The generalized procedure relates to a constructi on in category theory known as the left Kan extension. It admits of a great variety of applications, including enumerating cosets, computing certain colimits in the category of Sets, and enumerating the arrows in a category given by generators and relations. We begin by defining the notion of a left Kan extension, and giving a number of illustrativ e examples. We then provide a full specification of the procedure, fol lowed by its application in relation to each of the examples. Finally, we provide a formulation of the procedure in terms of graphs and pres entations of actions of graphs (automata) which is more convenient for theoretical purposes.