On Automorphisms of Categories of Universal Algebras

Given a variety V of universal algebras. A new approach is suggested to characterize algebraically automorphisms of the category of free V-algebras. It gives in many cases an answer to the problem set by the first of authors, if automorphisms of such a category are inner or not. This question is important for universal algebraic geometry [5, 9]. Most of results will actually be proved to hold for arbitrary categories with a represented forgetful functor. Mathematics Subject Classification 2000. 08C05, 08B20 INTRODUCTION It is a current opinion that the notions of an isomorphism and an automorphism of categories are not important. As far as the authors know there are no researches devoted to describing automorphisms of categories although this theme is very popular for the most of other algebraic structures. The first of authors set the problem to describe automorphisms of a category of free algebras of some given variety of universal algebras. It turns out that this problem is quite important for universal algebraic geometry [5, 9]. The most important case is, when all automorphisms of a category in question are inner or close to inner in a sense. Recall that an automorphism Φ of a category C is called inner if it is isomorphic to the identity functor Id in the category of all endofunctors of C. It means that for every object A of the given category there exists an isomorphism σA : A → Φ(A) such that for every morphism μ : A → B we have This research of the second author was partially supported by THE ISRAEL SCIENCE FOUNDATION founded by The Israel Academy of Sciences and Humanities Center of Excellence Program.