In logic, the false proposition, called falsehood or falsity, is the proposition which is always false.
The faleshood is commonly denoted $false$, $F$, $\bot$, or $0$.
In classical logic, there are two truth values: false and true. Classical logic is perfectly symmetric between falsehood and truth; see de Morgan duality.
In constructive logic, $false$ is the bottom element in the poset of truth values.
Constructive logic is still two-valued in the sense that any truth value is false if it is not true.
In terms of the internal logic of a topos (or other category), $false$ is the bottom element in the poset of subobjects of any given object (where each object corresponds to a context in the internal language).
However, not every topos is two-valued, so there may be other truth values besides $false$ and $true$.
In the archetypical topos Set, the terminal object is the singleton set $*$ (the point) and the poset of subobjects of that is classically $\{\emptyset \hookrightarrow *\}$. Then falsehood is the empty set $\emptyset$, seen as the empty subset of the point. (See Internal logic of Set for more details).
The same is true in the archetypical (∞,1)-topos ∞Grpd. From that perspective it makes good sense to think of
a set as a 0-truncated $\infty$-groupoid: a 0-groupoid;
a subsingleton set as a $(-1)$-truncated $\infty$-groupoid: a (−1)-groupoid.
In this sense, the object $false$ in Set or ∞Grpd may canonically be thought of as being the unique empty groupoid.
false, empty type