On automorphisms of categories with applications to universal algebraic geometry

Let \({\mathcal {V}}\) be a variety of algebras of some type \(\Omega \). An interest to describing automorphisms of the category \(\Theta ^0 ({\mathcal {V}})\) of finitely generated free \({\mathcal {V}}\)-algebras was inspired by 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. The method is to find all terms in the language of a given variety which determine such \(\Omega \)-algebras that are isomorphic to a given \(\Theta ^0 ({\mathcal {V}})\)-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 which works in more general setting. This method is based on two main theorems. The first of them gives a general description of automorphisms of categories which are supplied with a faithful representative functor into the category of sets. The second one shows how to obtain the full description of automorphisms of the category \(\Theta ^0 ({\mathcal {V}})\). This part of the paper 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). The last section contains some applications to universal algebraic geometry.