nLab
0-poset

A 0-poset is a truth value. Compare the concept of 11poset (a poset) and (1)(-1)-poset (which is trivial); compare also with (1)(-1)-category and 00-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 nn-categories, in particular the progression of nn-posets.

For example, there should be a 00-category of 00-posets; a 00-category is simply a set, and this set is the set of truth values, classically

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

Actually, we should expect the 00-category of 00-posets to be a 11-poset; this is simply a poset, and indeed truth values do form a poset (where \bot \leq \top).

If we equip the category of 00-posets with its monoidal cartesian structure (which is conjunction, the logical AND operation), then an \infty-category enriched over this should be a 11-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 (75.88.78.90)