Grobner Basis Techniques for Computing Actions of K-Categories

Abstract

This paper involves categories and computer science. The paper is motivated by a question which arises from two pieces of research. Firstly, the work of Brown and Heyworth which extends rewriting techniques to enable the computation of left Kan extensions over the category of sets. It is well known that left Kan extensions can be defined over categories other than Sets. Secondly, the `folklore' that rewriting theory is a special case of noncommutative Groebner basis theory. It is therefore natural to ask whether Groebner bases can provide a method for computing Kan extensions beyond the special case of rewriting. To answer this question completely, fully exploiting the computational power of Groebner basis techniques relating to Kan extensions is the ultimate aim. This paper provides a first step by showing how standard noncommutative Groebner basis procedures can be used to calculate left Kan extensions of K-category actions. In the final section of the paper a number of interesting problems arising from the work are identified.