The Todd-Coxeter procedure and left Kan extensions (Q1904139)

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 a great variety of applications, including enumerating cosets, computing certain colimits in the category of \({\mathcal S}ets\), 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. [See also the second and third author, in: Computational algebra and number theory, Math. Appl., Dordr. 325, 53-73 (1995; Zbl 0827.18001)].