A Boolean topos is a topos that is also a Boolean category.
There are several conditions on a topos that are necessary and sufficient to be Boolean:
Let $C$ be a Boolean pretopos, i.e. a pretopos in which is also a Boolean category, and let $Sh(C)$ be the classifying topos for $C$. Then it does not follow that $Sh(C)$ is Boolean. In fact, this is rarely the case, even if $C$ is the classifying pretopos for a theory in classical first-order logic. See Blass and Scedrov for a characterization of which classical first-order theories have Boolean classifying topoi – in particular, any such theory is $\aleph_0$-categorical. Nevertheless, as discussed below, Barr's theorem shows that any topos admits a surjection from a Boolean topos.
The internal logic of a Boolean topos with natural numbers object can serve as foundations for “ordinary” mathematics, except for that which relies on the axiom of choice. If you add the axiom of choice, then you get (an internal version of) ETCS; conversely, if you use an arbitrary topos, then you get constructive mathematics. (For some high-powered work, you may also need to add a version of the axiom of replacement or an axiom of Grothendieck universes.)
Every cartesian closed Boolean pretopos is in fact a topos. This is why ‘generalised predicativism’ (with function types but not power types) is necessarily a feature of constructive mathematics only.
For $\mathcal{E}$ a topos, then the following are equivalent:
$\mathcal{E}$ is a Boolean topos;
Every subtopos of $\mathcal{E}$ is Boolean.
Every subtopos of $\mathcal{E}$ is an open subtopos.
Every subtopos of $\mathcal{E}$ is a closed subtopos.
Let $j$ be a Lawvere-Tierney topology on $\mathcal{E}$. Then $\mathcal{E}_j$ is Boolean iff there exists a subterminal object $U$ such that $j$ is the largest topology such that $U\rightarrowtail 1$ is $j$-closed.
$\mathcal{E}$ is Boolean iff the only dense subtopos of $\mathcal{E}$ is $\mathcal{E}$ itself.
Proof. Suppose $\mathcal{E}$ is Boolean. $\mathcal{E}_{\neg\neg}=\mathcal{E}$ is the smallest dense subtopos (cf. double negation). Conservely, suppose $\mathcal{E}$ is not Boolean then $\mathcal{E}_{\neg\neg}$ is a second dense subtopos.
In a lattice of subtoposes the 2-valued Boolean toposes are the atoms.
See this proposition.
Let $\mathcal{E}$ be a topos. Then automorphisms of $\Omega$ correspond bijectively to closed Boolean subtoposes. The group operation on $Aut(\Omega)$ corresponds to symmetric difference of subtoposes.
This result appears in Johnstone (1979). (See also Johnstone (2002), A1.6.11 pp.66-67.)
With excluded middle in the meta-logic, every well-pointed topos is a Boolean topos. This includes Set and models of ETCS.
The topos of sheaves on a Boolean algebra $B$ for the coherent topology is Boolean if and only if the Boolean algebra is finite. Indeed, in $Sh(B)$ the sections of $\Omega$ over $b \in B$ are the ideals of $\downarrow b \leqslant B$, while those of $1+1$ are the principal ideals, and these coincide just when $B$ is finite.
The topos of canonical sheaves on a complete Boolean algebra is Boolean (note this includes the finite case above).
If $\mathcal{E}$ is any topos, the category of sheaves for the double-negation topology is a Boolean subtopos of $\mathcal{E}$.
Any topos satisfying the axiom of choice is Boolean. This result is due to R. Diaconescu (1975); see excluded middle for a brief discussion.
Barr's theorem implies that any topos $\mathcal{E}$ can be covered by a Boolean topos $\mathcal{F}$, in the sense of there being a surjective geometric morphism $f \colon \mathcal{F} \to \mathcal{E}$.
Boolean toposes are closely related to measurable spaces (e.g Jackson 06, Henry 14).
Andreas Blass, Andrej Scedrov, Boolean Classifying Topoi , JPAA 28 (1983) pp.15-30.
Radu Diaconescu, Axiom of Choice and Complementation , Trans. AMS 51 no.1 (1975) pp.176-178. (pdf)
Peter Johnstone, Automorphisms of $\Omega$ , Algebra Universalis 9 (1979) pp.1-7.
Peter Johnstone, Sketches of an Elephant vols. I,II, Oxford UP 2002. (A4.5.22, D3.4, D4.5)
On relation of Boolean toposes to measure theory:
Matthew Jackson, A sheaf-theoretic approach to measure theory, 2006. (pdf, d-scholarship:7348)
Simon Henry, Measure theory over boolean toposes, Mathematical Proceedings of the Cambridge Philosophical Society, 2016 (arXiv:1411.1605)
Asgar Jamneshan, Terence Tao, Foundational aspects of uncountable measure theory: Gelfand duality, Riesz representation, canonical models, and canonical disintegration (arXiv:2010.00681)
Last revised on February 9, 2023 at 20:18:07. See the history of this page for a list of all contributions to it.