Some remarks on autoequivalences of categories
نویسنده
چکیده
Prof. Boris I. Plotkin [1, 2] drew attention to the question when an equivalence between two categories is isomorphic as a functor to an isomorphism between them. It turns out that it is quite important for universal algebraical geometry and concerns mainly the categories Θ0(X) of free universal algebras of some variety Θ free generated by finite subsets of X. In the paper, a complete answer to the Plotkin’s question is given: there are no proper autoequivalences of the category Θ0(X). Also some connected problems are discussed. Introduction Prof. Boris I. Plotkin set a question arose by studying of the universal algebraical geometry [1, 2]. Without going into details, this question can be formulated in the following way. Let V be a variety of universal algebras. Consider the category Θ whose objects are all algebras from V and whose morphisms are all homomorphisms of them. Fix an infinite set X. Let Θ(X) be the full subcategory of Θ containing only free V−algebras over finite subsets of the set X. The question is: if there are autoequivalences of Θ(X) that are not isomorphic to any automorphism of this category? Below, we give a negative answer to this question, i. e. we show that every autoequivalence of the category Θ(X) is isomorphic to an automorphism of this category, and present some results concerning that theme. Properly to say Prof. B.I. Plotkin regards the mentioned question to be important for the universal algebraical geometry. The reasons for this opinion are the following ones. ∗This research is partially supported by THE ISRAEL SCIENCE FOUNDATION founded by The Israel Academy of Sciences and Humanities Center of Excellence Program.
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