truth value

Classically, a **truth value** is either $\top$ (True) or $\bot$ (False). (In constructive mathematics, this is not so simple, although it still holds that any truth value that is not true is false.)

More generally, a **truth value** in a topos $T$ is a morphism $1 \to \Omega$ (where $1$ is the terminal object and $\Omega$ is the subobject classifier) in $T$. By definition of $\Omega$, this is equivalent to an (equivalence class of) monomorphisms $U\hookrightarrow 1$. In a two-valued topos, it is again true that every truth value is either $\top$ or $\bottom$, while in a Boolean topos this is true in the internal logic.

Truth values form a poset (the **poset of truth values**) by declaring that $p$ precedes $q$ iff the conditional $p \to q$ is true. In a topos $T$, $p$ precedes $q$ if the corresponding subobject $P\hookrightarrow 1$ is contained in $Q\hookrightarrow 1$. Classically (or in a two-valued topos), one can write this poset as $\{\bot \to \top\}$.

The poset of truth values is a Heyting algebra. Classically (or internal to a Boolean topos), this poset is even a Boolean algebra. It is also a complete lattice; in fact, it can be characterised as the initial complete lattice. As a complete Heyting algebra, it is a frame, corresponding to the one-point locale.

When the set of truth values is equipped with the specialization topology, the result is Sierpinski space.

A truth value may be interpreted as a $0$-poset or as a $(-1)$-groupoid. It is also the best interpretation of the term ‘$(-1)$-category’, although this doesn't fit all the patterns of the periodic table.

homotopy level | n-truncation | homotopy theory | higher category theory | higher topos theory | homotopy type theory |
---|---|---|---|---|---|

h-level 0 | (-2)-truncated | contractible space | (-2)-groupoid | true/unit type/contractible type | |

h-level 1 | (-1)-truncated | contractible-if-inhabited | (-1)-groupoid/truth value | (0,1)-sheaf/ideal | mere proposition/h-proposition |

h-level 2 | 0-truncated | homotopy 0-type | 0-groupoid/set | sheaf | h-set |

h-level 3 | 1-truncated | homotopy 1-type | 1-groupoid/groupoid | (2,1)-sheaf/stack | h-groupoid |

h-level 4 | 2-truncated | homotopy 2-type | 2-groupoid | (3,1)-sheaf/2-stack | h-2-groupoid |

h-level 5 | 3-truncated | homotopy 3-type | 3-groupoid | (4,1)-sheaf/3-stack | h-3-groupoid |

h-level $n+2$ | $n$-truncated | homotopy n-type | n-groupoid | (n+1,1)-sheaf/n-stack | h-$n$-groupoid |

h-level $\infty$ | untruncated | homotopy type | ∞-groupoid | (∞,1)-sheaf/∞-stack | h-$\infty$-groupoid |

- Special properties of truth values:
- In synthetic topology with a dominance, some truth values are open.

Revised on October 19, 2016 18:06:44
by Mike Shulman
(192.195.154.58)