# Exponentiation and Euler measure

## 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 \( n \choose k \) 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
\( 2^{-1} = {1 \over 2} \quad\mathrm{and}\quad{{1/2} \choose {2}} =
-{1 \over 8} \).
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. (Other glimpses into
the theory may be found in [11].)