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Boolean classifying topoi

Authors
Journal
Journal of Pure and Applied Algebra
0022-4049
Publisher
Elsevier
Publication Date
Volume
28
Issue
1
Identifiers
DOI: 10.1016/0022-4049(83)90069-5
Keywords
  • Classifying Topos
  • Boolean Topos
  • Atomic Topos
  • Model-Complete Theory
  • X0-Categorical Theory
Disciplines
  • Mathematics

Abstract

Abstract Let λ be a finitary geometric theory and δ its classifying topos. We prove that δ is Boolean if and only if (1) every first-order formula in the language of λ is ⧸-provably equivalent to a geometric formula and (2) for any finite list of varibles, x, there are, up to ⧸-provable equivalence, only finitely many formulas, in the language of λ with free variables among x. We use this characterization to show that, when δ is Boolean, it is an atomic topos and can be viewed as a finite coproduct of topoi of continuous G-sets for topological groups G satisfying a certain finiteness condition.

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