nLab 0-poset

A 0-poset is a truth value. Compare the concept of $1$poset (a poset) and $(-1)$-poset (which is trivial); compare also with $(-1)$-category and $0$-groupoid, which mean the same thing for different reasons.

The point of 0-posets is that they complete some patterns in the periodic table of $n$-categories, in particular the progression of $n$-posets.

For example, there should be a $0$-category of $0$-posets; a $0$-category is simply a set, and this set is the set of truth values, classically

$$(-1)Pos := \{\bot, \top\} \,.$$

Actually, we should expect the $0$-category of $0$-posets to be a $1$-poset; this is simply a poset, and indeed truth values do form a poset (where $\bot \leq \top$).

If we equip the category of $0$-posets with its monoidal cartesian structure (which is conjunction, the logical AND operation), then an $\infty$-category enriched over this should be a $1$-poset; and indeed it is (up to equivalence of categories) a poset (although up to isomorphism only, a category enriched over truth values under conjunction is actually a set equipped with a preorder).

See (−1)-category for references on this sort of negative thinking.