nLab two-valued topos

Idea

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

Definition

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

Properties

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

• Since $\Omega$ classifies subobjects $X\subseteq 1$, these correspond precisely to global elements $1\overset{\chi_X}{\to}\Omega$ whence a topos $\mathcal{E}$ is two-valued precisely if the terminal object $1$ has exactly the two subobjects $0$ and $1$.

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

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

• Being two-valued is different from saying that $\Omega\cong 1\coprod 1$: The latter property characterizes Boolean toposes but a Boolean topos is not necessarily two-valued - the easiest consistent example possibly being $Set\times Set$ 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 $Set$, 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 $M$: Since the underlying set of $\Omega$ consists of all right ideals of $M$ with $m\in M$ acting on a right ideal $I$ by mapping it to $I\cdot m=\{x\in M|m\cdot x\in I\}\;,$ a global element picks out a right ideal $J$ that satisfies $J\cdot y= J$ for all $y\in M$, inheriting by equivariance the triviality of the action on $1$ with underlying set a singleton, which implies $J=\empty$ or $J=M$, the latter since $j\in J$ entails $1\in J\cdot j=J\;.$ Whence $Set^{M^{op}}$ is two-valued but it is a well known excercise that $Set^{M^{op}}$ is Boolean precisely iff $M$ is a group.

Related entries

• 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.