Remarks on piecewise-linear algebra

Abstract

This note studies some of the basic properties of the category whose objects are finite unions of (open and closed) and whose morphisms are (not necessarily continu- ous) piecewise-linear maps. Introduction* A function f: V —>W between real vector spaces is piecewise-linear (PL) if there exists a partition of V into polyhedra X€ (i.e., relative interiors of polyhedra) such that / is affine on each Xt. (As distinct to the case of PL-topology, no continuity is required of /.) Images and preimages under PL-maps give rise to finite unions of open polyhedra, or PL-seίs; conversely PL functions can be characterized by the fact that their graphs are PL-sets. This paper studies some basic algebraic properties of the category PL, proving in particular that it is an exact category, and in fact a pretopos. A classification is given for the isomorphism classes of objects of PL, in terms of a two-generator semiring. The first section recalls without proof some facts from polyhedral geometry needed in the paper. Except for the setting of a unified notation and for minor generalizations, the material there is well known. The second section defines PL maps and sets, and studies the category. The main results (leading to the classification theorem) are given in the last section. 1* Review of polyhedral geometry* The following conventions and definitions hold throughout. All vector spaces are finite- dimensional spaces over the reals R; a flat means an affine sub- manifold of some such space V, and the closed half-spaces associated to a linear /: V —> R and an r in R (or associated to the hyperplane {x\f(x) = r},) are the sets {x\f(x) ^ r} and {x\f(x) ^ r}. The cor- responding open half-spaces are obtained by using strict inequalities in the above. A half-line (closed or open) is the intersection of a line L in V with a (closed or open) half-space not containing L.