  We investigate similarities between the category of vector spaces and that of
polytopal algebras, containing the former as a full subcategory. In Section 2
we introduce the notion of a polytopal Picard group and show that it is trivial
for fields. The coincidence of this group with the ordinary Picard group for
general rings remains an open question. In Section 3 we survey some of the
previous results on the automorphism groups and retractions. These results
support a general conjecture proposed in Section 4 about the nature of
arbitrary homomorphisms of polytopal algebras. Thereafter a further
confirmation of this conjecture is presented by homomorphisms defined on
Veronese singularities.
  This is a continuation of the project started in our papers "Polytopal linear
groups" (J. Algebra 218 (1999), 715--737), "Polytopal linear retractions"
preprint, math.AG/0001049) and "Polyhedral algebras, arrangements of toric
varieties, and their groups" (preprint,
http://www.mathematik.uni-osnabrueck.de/K-theory/0232/index.html). The higher
$K$-theoretic aspects of polytopal linear objects will be treated in
"Polyhedral $K$-theory" (in preparation).
