Autoequivalences of categories of free semigroups and free inverse semigroups








G. I. Zhitomirski¹
Bar-Ilan University, Israel









It was very often mentioned that the notion of equivalence of two categories was more preferable than the notion of the automorphism for category theory itself and for most of its applications. But it is a very natural question in which cases there are no equivalences very different from automorphisms, i.e. they are not isomorphic to any automorphism. We call such autoequivalences proper autoequivalences. This question was set for the first time by B.I. Plotkin in connection with his studying the universal algebraical geometry. Plotkin's question concerned mainly some special categories namely the categories $\Theta \sp 0 (X)$ of free universal algebras of a variety $\Theta$ over finite subsets of a fixed infinite set $X$. But the answer led out this special case.


Theorem 1. Let ${\cal C}$ be an arbitrary small (it is enough for our aim) category. For every object $A$ of ${\cal C}$, let $[A]$ denote the set of all ${\cal C}$-objects, which are isomorphic to $A$. An autoequivalence $\alpha$ is isomorphic to an automorphism if and only if for every two objects $X$ and $Y$ such that $\alpha (X) = Y$ the sets $[X]$ and $[Y]$ are of equal cardinality.


A small category ${\cal C}$ has no proper autoequivalences if and only if for every automorphism $\gamma$ of a skeleton $Sk$ of this category and for every two objects $X$ and $Y$ of $Sk$ such that $\gamma (X)= Y$, the sets $[X]$ and $[Y]$ are of equal cardinality. This gives the following answer to Plotkin's question: in the categories of the kind $\Theta \sp 0 (X)$ there are no proper autoequivalences. The proof of Theorem 1 uses the Axiom of Choice but in some simple cases it is possible to remove references to this axiom, for example for the category $\Theta \sp 0 (X)$ if the set $X$ is denumerable.


Now let ${\cal C}$ be an arbitrary category supplied with a faithful functor $U:{\cal C} \to {Set}$ with the following condition:


(*) Let $A$ and $B$ be two objects of this category such that $U(A)$ and $U(B)$ have the same cardinality. Then for every bijection $\nu :U(A)\to U(B)$ there exists an unique object $C$ of ${\cal C}$ and an isomorphism $\nu \sp * :C\to B$ such that $U(C)= U(B))$ and $U(\nu \sp *) = \nu$. For such categories a sufficient condition is given for an autoequivalence $\pi$ to be isomorphic to an automorphism. In particular, this condition is satisfied if the functor $U$ is represented by an object $W$ such that $\pi (W)$ is isomorphic to $W$. Thus every full subcategory of $Set$ containing singleton does not contain any proper autoequivalences.


Let $\cal V$ be a variety of universal algebras and $\Theta$ be the corresponding category. Let $U: \Theta \to Set$ be the forgetting functor. It is clear that condition (*) is satisfied. Moreover, the functor $U$ is represented by the free algebra $W$ in ${\cal V }$ over an one-element set. Thus if $W$ is isomorphic to $\pi (W)$ for an autoequivalence $\pi$, then $\pi$ is isomorphic to an automorphism. Indeed, the same fact is valid for every full subcategory of $\Theta$ if it satisfies the condition (*) and contains a free algebra $W$ over a singleton. Hence we have


Theorem Let $\Theta$ be the category of all free semigroups, or of all free inverse semigroups or of all free groups. Then there are no proper autoequivalences in $\Theta$.





Sponsored in part by the FCT approved projects POCTI 32817/99 and POCTI/MAT/37670/2001 in participation with the European Community Fund FEDER and by FCT through Centro de Matemática da Universidade do Porto. Also sponsored in part by FCT, the Faculdade de Ciências da Universidade do Porto, Programa Operacional Ciência, Tecnologia, Inovação do Quadro Comunitário de Apoio III, and by Caixa Geral de Depósitos.

 

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