Term-Rewriting Systems and Categories of Effective Descent

Dali Zangurashvili

Razmadze Mathematical Institute 
Tbilisi, Republic of Georgia

Abstract

Grothendieck’s descent theory is a powerful tool for describing so-called locally trivial objects in various areas of algebra and geometry. The main notions of this theory are of a descent morphism and an effective descent morphism. Though effective descent morphisms were described in many categories from geometry, only little is known on the structure of such morphisms in categories dual to varieties of universal algebras. The problem of characterizing effective descent morphisms in these categories is important not only for Pure Mathematics (Category Theory, Modal Logic, Universal Algebra), but also for Computer Science (namely, in the theory of multi-agent systems). We give a sufficient condition for all descent morphisms of a particular category (that dual to a Mal’cev variety) to be effective, formulated in terms of term-rewriting systems.  This condition is satisfied, for instance, by the variety of quasigroups. Based on this sufficient condition and some other facts, we suggest a package of algorithms which can be added to Mathematica. Applying them, the problem of describing effective descent morphisms, in categories dual to these or other concrete varieties of universal algebras, will be solved.