A 0-poset is a truth value. Compare the concept of poset (a poset) and -poset (which is trivial); compare also with -category and -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 -categories, in particular the progression of -posets.
For example, there should be a -category of -posets; a -category is simply a set, and this set is the set of truth values, classically
Actually, we should expect the -category of -posets to be a -poset; this is simply a poset, and indeed truth values do form a poset (where ).
If we equip the category of -posets with its monoidal cartesian structure (which is conjunction, the logical AND operation), then an -category enriched over this should be a -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.
Revised on June 30, 2010 22:07:41
by Toby Bartels