When is a variety a topos ?
type: Item
instance of: scholarly article
title: When is a variety a topos ?
zbMATH Open document ID: 0599.18003
DOI: 10.1007/BF01188056
published in: Algebra Universalis
publication date: 1985-01-01T00:00:00Z
review text: 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.
Mathematics Subject Classification ID: 18B25
Mathematics Subject Classification ID: 18C10
Mathematics Subject Classification ID: 08C05
zbMATH DE Number: 3966363
zbMATH Keywords: categories of M-sets
zbMATH Keywords: categories of Jonsson-Tarski algebras
zbMATH Keywords: topos
zbMATH Keywords: algebraic theory
zbMATH Keywords: coproducts
author: Peter Johnstone
reviewed by: K. I. Rosenthal
MaRDI profile type: Publication
cites work: Sur quelques points d'algèbre homologique
cites work: On Two Properties of Free Algebras.
cites work: Q5599205
cites work: Q4123496
full work available at URL: https://doi.org/10.1007/bf01188056
OpenAlex ID: W2077980683
description: scientific article; zbMATH DE number 3966363