On automorphisms of categories

TL;DR

This paper introduces a new method to describe automorphisms of categories of finitely generated free algebras, overcoming previous restrictions, with applications to semigroups and modules over rings.

Contribution

A novel approach based on quasi-inner automorphisms that generalizes previous methods for describing automorphisms of algebraic categories.

Findings

Effective in the variety of all semigroups

Applied to modules over rings with unity

Produced new results for rings without zero divisors

Abstract

Let V be a variety of algebras of some type. An interest to describing automorphisms of the category C of finitely generated free V-algebras was inspired in connection with development of universal algebraic geometry founded by B. Plotkin. There are a lot of results on this subject. A common method of getting such results was suggested and applied by B. Plotkin and the author (B. Plotkin , G. Zhitomirski, On automorphisms of categories of free algebras of some varieties, J. Algebra 306(2) (2006) 344-367). The method is to find all terms in the language of a given variety which determine such V-algebras that are isomorphic to a given C-algebra and have the same underlying set with it. But this method can be applied only to automorphisms which take all objects to isomorphic ones. The aim of the present paper is to suggest another method that is free from the mentioned restriction. This method is based on two main theorems. Let C be a category supplied with a faithful representative functor to the category of sets. Theorem 1 gives a general description of automorphisms of C, using a new notion of a quasi-inner automorphism. Theorem 5 shows how to obtain the full description of automorphisms of the category of finitely generated free V-algebras. The investigation ends with two examples. The first of them shows the preference of our method in a known situation (the variety of all semigroups) and the second one demonstrates obtaining new results (the variety of all modules over arbitrary ring with unit and the consequence for the rings without zero divisors).