two-valued topos in nLab

Contents

Contents

Idea

A two-valued topos is a topos with exactly two truth values.

Definition

A topos ℰ\mathcal{E} is called two-valued if its subobject classifier Ω\Omega is two-valued, with precisely two global elements 1→trueΩ1\overset{true}{\to}\Omega and 1→falseΩ1\overset{false}{\to}\Omega.

Properties

• In particular, a two-valued topos ℰ\mathcal{E} is consistent i.e. ℰ≠1\mathcal{E}\neq 1.

• Since Ω\Omega classifies subobjects X⊆1X\subseteq 1, these correspond precisely to global elements 1→χ XΩ1\overset{\chi_X}{\to}\Omega whence a topos ℰ\mathcal{E} is two-valued precisely if the terminal object 11 has exactly the two subobjects 00 and 11.

• Two-valued toposes are connected i.e. 0≠10\neq 1 and U∐V≅1U\coprod V\cong 1 implies U≅1U\cong 1 and V≅0V\cong 0 or vice versa. This holds because UU, VV are disjoint subobjects of U∐V≅1U\coprod V\cong 1 but 11 has only the two trivial subobjects.

• Let 𝕋\mathbb{T} be a geometric theory over the signature Σ\Sigma. Then 𝕋\mathbb{T} is called complete if every geometric sentence φ\varphi over Σ\Sigma is 𝕋\mathbb{T}-provably equivalent to either ⊤\top or ⊥,\bottom\; , but not both. A geometric theory 𝕋\mathbb{T} is complete precisely iff its classifying topos Set[𝕋]Set[\mathbb{T}] is two-valued (Caramello 2012, remark 2.5).

• Being two-valued is different from saying that Ω≅1∐1\Omega\cong 1\coprod 1: The latter property characterizes Boolean toposes but a Boolean topos is not necessarily two-valued - the easiest consistent example possibly being Set×SetSet\times Set which is four-valued, more complicated ones often arising in intermediates steps in set-theoretic forcing (cf. continuum hypothesis). There exists a general filter quotient construction turning a Boolean topos into a two-valued Boolean topos (cf. Mac Lane-Moerdijk 1994, pp.256ff, 274).

Examples

• Though the forcing models resulting from the above mentioned filter-quotient construction provide a plethora of examples of Boolean two-valued toposes other than SetSet, two-valued toposes are by no means bound to be Boolean.

• In fact, there exists a rich supply of two-valued toposes that are not Boolean provided by the toposes of actions of monoids MM: Since the underlying set of Ω\Omega consists of all right ideals of MM with m∈Mm\in M acting on a right ideal II by mapping it to I⋅m={x∈M|m⋅x∈I},I\cdot m=\{x\in M|m\cdot x\in I\}\;, a global element picks out a right ideal JJ that satisfies J⋅y=JJ\cdot y= J for all y∈My\in M, inheriting by equivariance the triviality of the action on 11 with underlying set a singleton, which implies J=∅J=\empty or J=MJ=M, the latter since j∈Jj\in J entails 1∈J⋅j=J.1\in J\cdot j=J\;. Whence Set M opSet^{M^{op}} is two-valued but it is a well known excercise that Set M opSet^{M^{op}} is Boolean precisely iff MM is a group.

• two-valued object, two-valued logic

• well-pointed topos

• Boolean topos

• continuum hypothesis

• Deligne completeness theorem

References

• Olivia Caramello, Atomic toposes and countable categoricity , Appl. Cat. Struc. 20 no. 4 (2012) pp.379-391. (arXiv:0811.3547)

• Saunders Mac Lane, Ieke Moerdijk, Sheaves in Geometry and Logic , Springer Heidelberg 1994.