(-1)-poset

There is just one **$(-1)$-poset**, namely the point. Compare the concepts of $0$-poset (a truth value) and $1$-poset (a poset). Compare also with $(-2)$-category and $(-2)$-groupoid, which mean the same thing for their own reasons.

The point of $(-1)$-posets is that they complete some patterns in the periodic tables and complete the general concept of $n$-poset. For example, there should be a $0$-poset $(-1)\Pos$ of $(-1)$-posets; a $0$-poset is simply a truth value, and $(-1)\Pos$ is the true truth value.

As a category, $(-1)\Pos$ is a monoidal category in a unique way, and a category enriched over this should be (at least up to equivalence) a $0$-poset, which is a truth value; and indeed, a category enriched over $(-1)\Pos$ 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)