(0,1)-category

(0,1)-topos

# Contents

## Idea

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 leveln-truncationhomotopy theoryhigher category theoryhigher topos theoryhomotopy type theory
h-level 0(-2)-truncatedcontractible space(-2)-groupoidtrue/unit type/contractible type
h-level 1(-1)-truncated(-1)-groupoid/truth valuemere proposition, h-proposition
h-level 20-truncateddiscrete space0-groupoid/setsheafh-set
h-level 31-truncatedhomotopy 1-type1-groupoid/groupoid(2,1)-sheaf/stackh-groupoid
h-level 42-truncatedhomotopy 2-type2-groupoidh-2-groupoid
h-level 53-truncatedhomotopy 3-type3-groupoidh-3-groupoid
h-level $n+2$$n$-truncatedhomotopy n-typen-groupoidh-$n$-groupoid
h-level $\infty$untruncatedhomotopy type∞-groupoid(∞,1)-sheaf/∞-stackh-$\infty$-groupoid

Revised on June 22, 2014 17:33:24 by Toby Bartels (98.16.175.187)