Article,
Exponentiation and Euler measure
(2002)cite arxiv:math/0204009Comment: 15 pages; to appear in Algebra Universalis.
Abstract
Two of the pillars of combinatorics are the notion of choosing an arbitrary
subset of a set with $n$ elements (which can be done in $2^n$ ways), and the
notion of choosing a $k$-element subset of a set with $n$ elements (which can
be done in $n k$ ways). In this article I sketch the beginnings of a
theory that would import these notions into the category of hedral sets in the
sense of Morelli and the category of polyhedral sets in the sense of Schanuel.
Both of these theories can be viewed as extensions of the theory of finite sets
and mappings between finite sets, with the concept of cardinality being
replaced by the more general notion of Euler measure (sometimes called
combinatorial Euler characteristic). I prove a ``functoriality'' theorem
(Theorem 1) for subset-selection in the context of polyhedral sets, which
provides quasi-combinatorial interpretations of assertions such as $2^-1 =
\frac12$ and $1/2 2 = -\frac18$. Furthermore, the operation of
forming a power set can be viewed as a special case of the operation of forming
the set of all mappings from one set to another; I conclude the article with a
polyhedral analogue of the set of all mappings between two finite sets, and a
restrictive but suggestive result (Theorem 2) that offers a hint of what a
general theory of exponentiation in the polyhedral category might look like.
