nLab
(-1)-poset

There is just one (1)(-1)-poset, namely the point. Compare the concepts of 00-poset (a truth value) and 11-poset (a poset). Compare also with (2)(-2)-category and (2)(-2)-groupoid, which mean the same thing for their own reasons.

The point of (1)(-1)-posets is that they complete some patterns in the periodic tables and complete the general concept of nn-poset. For example, there should be a 00-poset (1)Pos(-1)\Pos of (1)(-1)-posets; a 00-poset is simply a truth value, and (1)Pos(-1)\Pos is the true truth value.

As a category, (1)Pos(-1)\Pos is a monoidal category in a unique way, and a category enriched over this should be (at least up to equivalence) a 00-poset, which is a truth value; and indeed, a category enriched over (1)Pos(-1)\Pos is a category in which any two objects are isomorphic in a unique way, which is equivalent to a truth value.

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

Revised on June 30, 2010 22:03:59 by Toby Bartels (75.88.78.90)