When is a variety a topos ?
Besides categories of M-sets, where M is a monoid, the categories of Jonsson-Tarski algebras seem to be the only other known examples of varieties, which are also topoi. The (binary) Jonsson-Tarski variety was first observed to be a topos by P. Freyd. This example was generalized, first by the reviewer and then by Z. Barel. In this article, the question of when T-Alg, the category of T-algebras for an algebraic theory T, is a topos is given a syntactic answer, motivated in part by consideration of the Jonsson-Tarski examples. If p is an n-ary operation of T (n could be an infinite cardinal), then a unary operation u is called p-unary iff \(u(p(x_ 1,...,x_ n))\) is a word in just one of the variables \(x_ 1,...,x_ n\). p is called sufficiently unary iff there exists an m-ary operation q and m unary operations \(u_ 1,...,u_ m\) with \(q(u_ 1(y),...,u_ m(y))=y\) an equation of T. The result is that for a non-degenerate algebraic theory T, T-Alg is a topos iff every operation of T is sufficiently unary and T has no pseudo-constants (unary operations u for which \(u(x)=u(y)\) is an equation). The necessity of these two conditions comes from consideration of coproducts. Disjointness of coproducts together with the strictness of the initial object implies that there are no pseudo-constants, while universality of coproducts forces every operation to be sufficiently unary.
scientific article; zbMATH DE number 3966363