The Todd-Coxeter Procedure and Left Kan Extensions

We introduce a generalization of the Todd-Coxeter procedure for the enumeration of cosets. The generalized procedure relates to a construction 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 illustrative examples. We then provide a full specification of the procedure, followed by its application in relation to each of the examples. Finally, we provide a formulation of the procedure in terms of graphs and presentations of actions of graphs (automata) which is more convenient for theoretical purposes