Effective codescent morphisms in varieties of universal algebras with the amalgamation property

The notion of an effective descent morphism is one of the main notions of A. Grothendieck’s descent theory. The problem of characterizing such morphisms (called a descent problem) in varieties of universal algebras is simple and well-known. In contrast to this case, for the dual category of a variety of universal algebras, very little is known. The first result of A. Grothendieck, still used in commutative algebra and algebraic geometry, can be seen as a partial solution to the descent problem for the dual category of commutative rings. The complete solution was independently obtained by A. Joyal and M. Tierney, and B. Mesablishvili later. This talk is devoted to the descent problem in general dual-algebraic categories, which was posed by G. Janelidze. We deal with the case of categories with the amalgamation property (which is satisfied by a number of well known categories, but not by that of commutative rings). In [3], we gave the criterion for a morphism of a category with pushouts and equalizers to be a codescent morphism (we use the term “codescent” for descent in dual categories). In [4], we reduced the codescent problem to the simpler one, replacing arbitrary pushouts in codescent data by pushouts of monomorphisms. Applying this, we obtained the characterization of effective codescent morphisms in a number of categories of topological nature: topological spaces, compact Hausdorff topological spaces, normal topological spaces, Banach spaces, and some others. Moreover, we found the sufficient condition formulated in syntactical form for all codescent morphism of a variety of universal algebras to be effective [5]; with its aid the complete answer to the codescent problem in the categories of groups, loops, quasigroups was given. In [1], we employed the tools of the term rewriting systems theory to the codescent problem, and solved the codescent problem for the varieties of Mal’tsev algebras, idempotent quasigroups, unipotent quasigroups, and some others. Recently, appying the results of this paper, we solved the codescent problem in the varieties of n-loops and n-quasigroups [6], and ternary rings [2]. Note that a ternary ring is not a ring in the traditional sense. However, the category of ternary rings contains the category of traditional rings with unit as a full reflective subcategory. Our results imply that the class of morphisms between commutative rings which are effective codescent in the category of ternary rings coincides with that of monomorphisms satisfying the traditional ring-theoretic ideal extension property. Note that the problem whether the latter class of monomorphisms coincides with that of monomorphisms satisfying the above-mentioned Joyal-Tierney’s condition was of interest for certain purely ring-theoretic reasons in the past century, and the negative answer was given independently by several authors. The author gratefully acknowledges the financial support of Shota Rustaveli National Science Foundation of Georgia (FR-24-8249).